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Chapter 5

Eigenvalues and Eigenvectors

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5.1 · Chapter Problems

5.2 · Supplementary Problems

Section 5.1Chapter Problems[86/100%]

Problem 5.1.1

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

Find the eigenvalues and normalized eigenvectors of the $2 \times 2$ matrix

A = \begin{pmatrix} \sin (\theta ) & \cos (\theta ) \\ -\cos (\theta ) & \sin (\theta ) \end{pmatrix}.

Are the eigenvectors orthogonal to each other?

(ii)

Find the eigenvalues and normalized eigenvectors of the $2 \times 2$ matrix

\begin{pmatrix} \cos (\theta ) & -e^{i\phi }\sin (\theta ) \\ e^{-i\phi }\sin (\theta ) & \cos (\theta ) \end{pmatrix}.

(iii)

Consider the normalized vector in $\mathbb {C}^{3}$

\overrightarrow {n} = \begin{pmatrix} \sin (\theta )\cos (\phi ) \\ \sin (\theta )\sin (\phi ) \\ \cos (\theta ) \end{pmatrix}.

Calculate the $2 \times 2$ matrix $U(\theta ,\phi ) = \overrightarrow {n} \cdot \overrightarrow {\sigma } \equiv n_{1}\sigma_{1} + n_{2}\sigma_{2} + n_{3}\sigma_{3}$ , where $\sigma_{1}, \sigma_{2}, \sigma_{3}$ are the Pauli spin matrices. Is the matrix $U(\theta ,\phi )$ unitary? Find the trace and the determinant. Is the matrix $U(\theta ,\phi )$ hermitian? Find the eigenvalues and normalized eigenvectors of $U(\theta ,\phi )$ .

Problem 5.1.2

Find all solutions of the linear equation

\begin{pmatrix} \cos (\theta ) & -\sin (\theta ) \\ -\sin (\theta ) & -\cos (\theta ) \end{pmatrix}\overrightarrow {x} = \overrightarrow {x}, \qquad \theta \in \mathbb {R} \tag {1}

with the condition that $\overrightarrow {x} \in \mathbb {R}^{2}$ and $\overrightarrow {x}^{T}\overrightarrow {x}=1$ , i.e. the vector $\overrightarrow {x}$ must be normalized. What type of equation is (1)?

Origin: Stated according to the original source

Problem 5.1.3

(i)(ii)

Origin: Stated according to the original source

(i)

Let $\epsilon \in \mathbb {R}$ . Find the eigenvalues and eigenvectors of the $2 \times 2$ matrix

A(\epsilon ) = \frac{1}{\sqrt{1+\epsilon ^{2}}}\begin{pmatrix} \epsilon & 1 \\ 1 & -\epsilon \end{pmatrix}.

For $\epsilon =0$ we obtain the Pauli spin matrix $\sigma_{1}$ , for $\epsilon =1$ we have the Hadamard matrix and for $\epsilon \rightarrow \infty$ we obtain the Pauli spin matrix $\sigma_{3}$ .

(ii)

Let $\epsilon \in \mathbb {R}$ . Find the eigenvalues and eigenvectors of the $2 \times 2$ matrix

B(\epsilon ) = \begin{pmatrix} \tanh (\epsilon ) & 1/\cosh (\epsilon ) \\ 1/\cosh (\epsilon ) & -\tanh (\epsilon ) \end{pmatrix}.

The matrices $A(\epsilon )$ and $B(\epsilon )$ are connected via the invertible transformation $\epsilon \rightarrow \sinh (\epsilon )$ .

Problem 5.1.4

Let $A$ be a $2 \times 2$ matrix. Assume that $\operatorname {det}\left(A\right)=0$ and $\operatorname {tr}(A)=0$ . What can be said about the eigenvalues of $A$ ?

Origin: Stated according to the original source

Problem 5.1.5

Find all $2 \times 2$ matrices $A$ over $\mathbb {C}$ which admit the normalized eigenvectors

\overrightarrow {v}_{1} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}, \qquad \overrightarrow {v}_{2} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -1 \end{pmatrix}

with the corresponding eigenvalues $\lambda_{1}$ and $\lambda_{2}$ .

Origin: Stated according to the original source

Problem 5.1.6

Find the eigenvalues and normalized eigenvectors of the $2 \times 2$ matrix

M(\theta ) = \begin{pmatrix} \sin (\theta ) & \cos (\theta ) \\ \cos (\theta ) & -\sin (\theta ) \end{pmatrix}.

Extend the problem to the matrix

\begin{pmatrix} \sin (\theta ) & e^{i\phi }\cos (\theta ) \\ e^{-i\phi }\cos (\theta ) & -\sin (\theta ) \end{pmatrix}.

This matrix contains the Pauli spin matrices $\sigma_{1}, \sigma_{2}, \sigma_{3}$ for $\theta =0, \phi =0$ ; $\theta =0, \phi =3\pi /2$ ; $\theta =\pi /2, \phi =0$ .

Origin: Stated according to the original source

Problem 5.1.7

(i)(ii)

Let $\mu_{1}, \mu_{2} \in \mathbb {R}$ . Consider the $2 \times 2$ matrix

A(\mu _{1}^{2},\mu _{2}^{2}) = \begin{pmatrix} \mu _{1}^{2}\cos ^{2}(\theta )+\mu _{2}^{2}\sin ^{2}(\theta ) & (\mu _{2}^{2}-\mu _{1}^{2})\cos (\theta )\sin (\theta ) \\ (\mu _{2}^{2}-\mu _{1}^{2})\sin (\theta )\cos (\theta ) & \mu _{1}^{2}\sin ^{2}(\theta )+\mu _{2}^{2}\cos ^{2}(\theta ) \end{pmatrix}.

Origin: Stated according to the original source

(i)

Find the trace and determinant of $A(\mu_{1}^{2},\mu_{2}^{2})$ .

(ii)

Find the eigenvalues of $A(\mu_{1}^{2},\mu_{2}^{2})$ .

Remark.

Origin: Stated according to the original source

Any positive semidefinite $2 \times 2$ matrix over $\mathbb {R}$ can be written in this form. Try the $2 \times 2$ matrix with all entries $+1$ .

Problem 5.1.8

(i)(ii)

Origin: Stated according to the original source

(i)

Let $\alpha \in \mathbb {R}$ . Consider the matrices

A(\alpha ) = \begin{pmatrix} \cos (\alpha ) & -\sin (\alpha ) \\ \sin (\alpha ) & \cos (\alpha ) \end{pmatrix}, \qquad B(\alpha ) = \begin{pmatrix} \cos (\alpha ) & \sin (\alpha ) \\ \sin (\alpha ) & -\cos (\alpha ) \end{pmatrix}.

Find the trace and determinant of these matrices. Show that for the matrix $A(\alpha )$ the eigenvalues depend on $\alpha$ but the eigenvectors do not. Show that for the matrix $B(\alpha )$ the eigenvalues do not depend on $\alpha$ but the eigenvectors do.

(ii)

Let $\alpha \in \mathbb {R}$ . Consider the matrices

C(\alpha ) = \begin{pmatrix} \cosh (\alpha ) & \sinh (\alpha ) \\ \sinh (\alpha ) & \cosh (\alpha ) \end{pmatrix}, \qquad D(\alpha ) = \begin{pmatrix} \cosh (\alpha ) & -\sinh (\alpha ) \\ \sinh (\alpha ) & -\cosh (\alpha ) \end{pmatrix}.

Find the trace and determinant of these matrices. Show that for the matrix $C(\alpha )$ the eigenvalues depend on $\alpha$ but the eigenvectors do not. Show that for the matrix $D(\alpha )$ the eigenvalues do not depend on $\alpha$ but the eigenvectors do.

Problem 5.1.9

Let $a_{11}, a_{22} \in \mathbb {R}$ and $a_{12} \in \mathbb {C}$ . Consider the $2 \times 2$ hermitian matrix

H = \begin{pmatrix} a_{11} & a_{12} \\ \overline{a}_{12} & a_{22} \end{pmatrix}

with the real eigenvalues $\lambda_{+}$ and $\lambda_{-}$ . What conditions are imposed on the matrix elements of $H$ if $\lambda_{+}=\lambda_{-}$ ?

Origin: Stated according to the original source

Problem 5.1.10

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

Let $A^{T}=(1/2,1/2)^{T}$ . Find the eigenvalues of the $2 \times 2$ matrix $AA^{T}$ and the $1 \times 1$ matrix $A^{T}A$ .

(ii)

Let $\overrightarrow {x}$ be a nonzero column vector in $\mathbb {R}^{n}$ . Then $\overrightarrow {x}\overrightarrow {x}^{T}$ is an $n \times n$ matrix and $\overrightarrow {x}^{T}\overrightarrow {x}$ is a real number. Show that $\overrightarrow {x}^{T}\overrightarrow {x}$ is an eigenvalue of $\overrightarrow {x}\overrightarrow {x}^{T}$ and $\overrightarrow {x}$ is the corresponding eigenvector.

(iii)

Let $\overrightarrow {x}$ be a nonzero column vector in $\mathbb {R}^{n}$ and $n \geq 2$ . Consider the $n \times n$ matrix $\overrightarrow {x}\overrightarrow {x}^{T}$ . Find one nonzero eigenvalue and the corresponding eigenvector of this matrix.

Problem 5.1.11

Consider the Pauli spin matrix $\sigma_{1}$ with the eigenvalues $+1$ and $-1$ and the corresponding normalized eigenvectors

\overrightarrow {v}_{1} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}, \qquad \overrightarrow {v}_{2} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -1 \end{pmatrix}.

With the two normalized eigenvectors we form the unitary matrix

U = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \equiv \frac{1}{\sqrt{2}}(\sigma _{3}+\sigma _{1}).

Find the eigenvalues and normalized eigenvectors of $U$ .

Origin: Stated according to the original source

Problem 5.1.12

Let $\sigma_{1}, \sigma_{2}, \sigma_{3}$ be the Pauli spin matrices. Find the eigenvalues and normalized eigenvectors of the $2 \times 2$ nonnormal matrix

\sigma _{3}+i\sigma _{1} = \begin{pmatrix} 1 & i \\ i & -1 \end{pmatrix}.

Origin: Stated according to the original source

Problem 5.1.13

(i)(ii)

Origin: Stated according to the original source

(i)

Find all $2 \times 2$ matrices over $\mathbb {R}$ that admit only one eigenvector.

(ii)

Consider the $2 \times 2$ matrix

A(\epsilon ) = \begin{pmatrix} 1 & 0 \\ \epsilon & 1 \end{pmatrix}, \qquad \epsilon \in \mathbb {R}.

Can one find a condition on the parameter $\epsilon$ so that $A(\epsilon )$ has only one eigenvector?

Problem 5.1.14

Find all $2 \times 2$ matrices over $\mathbb {R}$ which commute with

A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}.

What is the relation between the eigenvectors of these matrices?

Origin: Stated according to the original source

Problem 5.1.15

(i)(ii)

Origin: Stated according to the original source

(i)

Let $A$ , $B$ be $2 \times 2$ matrices over $\mathbb {R}$ and vectors $\overrightarrow {x}$ , $\overrightarrow {y}$ in $\mathbb {R}^{2}$ such that $A\overrightarrow {x}=\overrightarrow {y}$ , $B\overrightarrow {y}=\overrightarrow {x}$ , $\overrightarrow {x}^{T}\overrightarrow {y}=0$ and $\overrightarrow {x}^{T}\overrightarrow {x}=1$ , $\overrightarrow {y}^{T}\overrightarrow {y}=1$ . Show that $AB$ and $BA$ have an eigenvalue $+1$ .

(ii)

Find all $2 \times 2$ matrices $A$ , $B$ which satisfy the conditions given in (i). Use

\overrightarrow {x} = \begin{pmatrix} \cos (\alpha ) \\ \sin (\alpha ) \end{pmatrix}, \qquad \overrightarrow {y} = \begin{pmatrix} -\sin (\alpha ) \\ \cos (\alpha ) \end{pmatrix}.

Problem 5.1.16

Let $A=(a_{jk})$ be a normal nonsymmetric $3 \times 3$ matrix over the real numbers. Show that

\overrightarrow {a} = \begin{pmatrix} a_{1} \\ a_{2} \\ a_{3} \end{pmatrix} = \begin{pmatrix} a_{23}-a_{32} \\ a_{31}-a_{13} \\ a_{12}-a_{21} \end{pmatrix}

is an eigenvector of $A$ .

Origin: Stated according to the original source

Problem 5.1.17

Let $c \in \mathbb {R}$ and consider the symmetric $3 \times 3$ matrix

A(c) = \begin{pmatrix} c & 1 & 0 \\ 1 & c & 1 \\ 0 & 1 & c \end{pmatrix}.

Show that $c$ is an eigenvalue of $A$ and find the corresponding eigenvector. Find the two other eigenvalues and normalized eigenvectors.

Origin: Stated according to the original source

Problem 5.1.18

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

Let $\alpha \in \mathbb {R}$ . Find the eigenvalues of

A(\alpha ) = \begin{pmatrix} 0 & \alpha & 0 \\ \alpha & 0 & \alpha \\ 0 & \alpha & 0 \end{pmatrix}.

Do the eigenvalues cross as a function of $\alpha$ ?

(ii)

Let $\alpha \in \mathbb {R}$ . Find the eigenvalues and eigenvectors of the $3 \times 3$ matrix

B(\alpha ) = \begin{pmatrix} -1 & \alpha & 0 \\ \alpha & 0 & \alpha \\ 0 & \alpha & 1 \end{pmatrix}.

Discuss the dependence of the eigenvalues and eigenvectors on $\alpha$ .

(iii)

Let $\alpha \in \mathbb {R}$ . Find the eigenvalues of the $4 \times 4$ symmetric matrix

C(\alpha ) = \begin{pmatrix} -1 & \alpha & 0 & 0 \\ \alpha & -1/2 & \alpha & 0 \\ 0 & \alpha & 1/2 & \alpha \\ 0 & 0 & \alpha & 1 \end{pmatrix}.

Discuss the eigenvalues $\lambda_{j}(\alpha )$ as functions of $\alpha$ .

Problem 5.1.19

Consider the following $3 \times 3$ matrix $A$ and vector $\overrightarrow {v}$ in $\mathbb {R}^{3}$

A = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}, \qquad \overrightarrow {v} = \begin{pmatrix} \sin (\alpha ) \\ \sin (2\alpha ) \\ \sin (3\alpha ) \end{pmatrix}

where $\alpha \in \mathbb {R}$ and $\alpha \neq n\pi$ with $n \in \mathbb {Z}$ . Show that using this vector we can find the eigenvalues and eigenvectors of $A$ . Start of with $A\overrightarrow {v}=\lambda \overrightarrow {v}$ .

Origin: Stated according to the original source

Problem 5.1.20

Consider the symmetric matrix over $\mathbb {R}$

A = \begin{pmatrix} 2 & 1 & -1 \\ 1 & 1 & 0 \\ -1 & 0 & 1 \end{pmatrix}.

Find an invertible matrix $B$ such that $B^{-1}AB$ is a diagonal matrix. Construct the matrix $B$ from the normalized eigenvectors of $A$ .

Origin: Stated according to the original source

Problem 5.1.21

Find the eigenvalues of the $3 \times 3$ symmetric matrix over $\mathbb {R}$

A = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix}

using the trace and the determinant of the matrix and the information that two eigenvalues are the same.

Origin: Stated according to the original source

Problem 5.1.22

Consider the symmetric matrix

A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{12} & a_{22} & a_{23} \\ a_{13} & a_{23} & a_{33} \end{pmatrix}

over $\mathbb {R}$ . Write down the characteristic polynomial $\operatorname {det}\left(\lambda I_{3}-A\right)$ and express it using the trace and determinant of $A$ .

Origin: Stated according to the original source

Problem 5.1.23

Consider the quadratic form

7x_{1}^{2}+6x_{2}^{2}+5x_{3}^{2}-4x_{1}x_{2}-4x_{2}x_{3}+14x_{1}-8x_{2}+10x_{3}+6 = 0.

Write this equation in matrix form and find the eigenvalues and normalized eigenvectors of the $3 \times 3$ matrix.

Origin: Stated according to the original source

Problem 5.1.24

(i)(ii)(iii)(iv)

Origin: Stated according to the original source

(i)

Let $H$ be an $n \times n$ matrix over $\mathbb {C}$ . Assume that $H$ is hermitian and unitary. What can be said about the eigenvalues of $H$ ?

(ii)

An $n \times n$ matrix $A$ such that $A^{2}=A$ is called idempotent. What can be said about the eigenvalues of such a matrix?

(iii)

An $n \times n$ matrix $A$ for which $A^{p}=0_{n}$ , where $p$ is a positive integer, is called nilpotent. What can be said about the eigenvalues of such a matrix?

(iv)

An $n \times n$ matrix $A$ such that $A^{2}=I_{n}$ is called involutory. What can be said about the eigenvalues of such a matrix?

Problem 5.1.25

(i)(ii)

Origin: Stated according to the original source

(i)

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . Show that the eigenvectors corresponding to distinct eigenvalues are linearly independent.

(ii)

Show that eigenvectors of a normal matrix $M$ corresponding to distinct eigenvalues are orthogonal.

Problem 5.1.26

Let $A$ be an $n \times n$ hermitian matrix, i.e. $A=A^{*}$ . Assume that all $n$ eigenvalues are different. Then the normalized eigenvectors $\left\{ \overrightarrow {v}_{j} : j=1,\ldots ,n\right\}$ form an orthonormal basis in $\mathbb {C}^{n}$ . Consider

\beta := (A\overrightarrow {x}-\mu \overrightarrow {x}, A\overrightarrow {x}-\nu \overrightarrow {x}) \equiv (A\overrightarrow {x}-\mu \overrightarrow {x})^{*}(A\overrightarrow {x}-\nu \overrightarrow {x})

where $(\, , \, )$ denotes the scalar product in $\mathbb {C}^{n}$ and $\mu , \nu$ are real constants with $\mu < \nu$ . Show that if no eigenvalue lies between $\mu$ and $\nu$ , then $\beta \geq 0$ .

Origin: Stated according to the original source

Problem 5.1.27

(i)(ii)

Let $A$ be an arbitrary $n \times n$ matrix over $\mathbb {C}$ . Let

H := \frac{A+A^{*}}{2}, \qquad S := \frac{A-A^{*}}{2i}.

Let $\lambda$ be an eigenvalue of $A$ and $\overrightarrow {x}$ be the corresponding normalized eigenvector (column vector).

Origin: Stated according to the original source

(i)

Show that $\lambda = \overrightarrow {x}^{*}H\overrightarrow {x}+i\overrightarrow {x}^{*}S\overrightarrow {x}$ .

(ii)

Show that the real part $\lambda_{r}$ of the eigenvalue $\lambda$ is given by $\lambda_{r}=\overrightarrow {x}^{*}H\overrightarrow {x}$ and the imaginary part $\lambda_{i}$ is given by $\lambda_{i}=\overrightarrow {x}^{*}S\overrightarrow {x}$ .

Problem 5.1.28

Let $\lambda_{1}, \lambda_{2}$ and $\lambda_{3}$ be the eigenvalues of the matrix

A = \begin{pmatrix} 0 & 1 & 2 \\ 0 & 0 & 1 \\ 2 & 2 & 1 \end{pmatrix}.

Find $\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}$ without calculating the eigenvalues of $A$ or $A^{2}$ .

Origin: Stated according to the original source

Problem 5.1.29

(i)(ii)(iii)

Consider the column vectors $\overrightarrow {u}$ and $\overrightarrow {v}$ in $\mathbb {R}^{n}$

\overrightarrow {u} = \begin{pmatrix} \cos (\theta ) \\ \cos (2\theta ) \\ \vdots \\ \cos (n\theta ) \end{pmatrix}, \qquad \overrightarrow {v} = \begin{pmatrix} \sin (\theta ) \\ \sin (2\theta ) \\ \vdots \\ \sin (n\theta ) \end{pmatrix}

where $n \geq 3$ and $\theta =2\pi /n$ .

Origin: Stated according to the original source

(i)

Calculate $\overrightarrow {u}^{T}\overrightarrow {u}+\overrightarrow {v}^{T}\overrightarrow {v}$ .

(ii)

Calculate $\overrightarrow {u}^{T}\overrightarrow {u}-\overrightarrow {v}^{T}\overrightarrow {v}+2i\overrightarrow {u}^{T}\overrightarrow {v}$ .

(iii)

Calculate the matrix $A=\overrightarrow {u}\overrightarrow {u}^{T}-\overrightarrow {v}\overrightarrow {v}^{T}$ , $A\overrightarrow {u}$ and $A\overrightarrow {v}$ . Discuss.

Problem 5.1.30

(i)(ii)

Origin: Stated according to the original source

(i)

Let $\overrightarrow {u}$ be a nonzero column vector in $\mathbb {R}^{n}$ . Consider the $n \times n$ matrix

A = \overrightarrow {u}\overrightarrow {u}^{T} - \overrightarrow {u}^{T}\overrightarrow {u}I_{n}.

Is $\overrightarrow {u}$ an eigenvector of this matrix? If so what is the eigenvalue?

(ii)

Let $\overrightarrow {u}, \overrightarrow {v}$ be nonzero column vectors in $\mathbb {R}^{n}$ and $\overrightarrow {u} \neq \overrightarrow {v}$ . Consider the $n \times n$ matrix $A$ over $\mathbb {R}$

A = \overrightarrow {u}\overrightarrow {u}^{T} + \overrightarrow {u}\overrightarrow {v}^{T} - \overrightarrow {v}\overrightarrow {u}^{T} - \overrightarrow {v}\overrightarrow {v}^{T}.

Find the nonzero eigenvalues of $A$ and the corresponding eigenvector. Note that $\overrightarrow {u}^{T}\overrightarrow {v}=\overrightarrow {v}^{T}\overrightarrow {u}$ .

Problem 5.1.31

(i)(ii)(iii)

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . We define

r_{j} := \sum _{\substack {k=1 \\ k \neq j}}^{n} \left|a_{jk}\right|, \qquad j=1,\ldots ,n.

Origin: Stated according to the original source

(i)

Show that each eigenvalue $\lambda$ of $A$ satisfies at least one of the following inequalities

\left|\lambda -a_{jj}\right| \leq r_{j}, \qquad j=1,\ldots ,n.

In other words show that all eigenvalues of $A$ can be found in the union of disks

\left\{ z : \left|z-a_{jj}\right| \leq r_{j}, \; j=1,\ldots ,n\right\} .

This is Gersgorin disk theorem.

(ii)

Apply this theorem to the $2 \times 2$ unitary matrix

A = \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}.

(iii)

Apply this theorem to the $3 \times 3$ nonnormal matrix

B = \begin{pmatrix} 1 & 2 & 3 \\ 3 & 4 & 9 \\ 1 & 1 & 1 \end{pmatrix}.

Problem 5.1.32

Let $A$ be a normal matrix over $\mathbb {C}$ , i.e. $A^{*}A=AA^{*}$ . Show that if $\overrightarrow {x}$ is an eigenvector of $A$ with eigenvalue $\lambda$ , then $\overrightarrow {x}$ is an eigenvector of $A^{*}$ with eigenvalue $\overline{\lambda }$ .

Origin: Stated according to the original source

Problem 5.1.33

(i)(ii)

Origin: Stated according to the original source

(i)

Show that an $n \times n$ matrix $A$ is singular if and only if at least one eigenvalue is $0$ .

(ii)

Let $B$ be an invertible $n \times n$ matrix. Show that if $\overrightarrow {x}$ is an eigenvector of $B$ with eigenvalue $\lambda$ , then $\overrightarrow {x}$ is an eigenvector of $B^{-1}$ with eigenvalue $\lambda^{-1}$ .

Problem 5.1.34

Let $A$ be an $n \times n$ matrix over $\mathbb {R}$ . Show that $A$ and $A^{T}$ have the same eigenvalues.

Origin: Stated according to the original source

Problem 5.1.35

Let $A$ be an $n \times n$ real symmetric matrix and

Q(\overrightarrow {x}) := \overrightarrow {x}^{T}A\overrightarrow {x}.

The following statements hold (maximum principle)

$\lambda_{1} = \max_{\left\| \overrightarrow {x}\right\| =1}(Q(\overrightarrow {x})) = Q(\overrightarrow {x}_{1})$ is the largest eigenvalue of the matrix $A$ and $\overrightarrow {x}_{1}$ is the eigenvector corresponding to eigenvalue $\lambda_{1}$ .
(inductive statement). Let $\lambda_{k} = \max Q(\overrightarrow {x})$ subject to the constraints
1. $\overrightarrow {x}^{T}\overrightarrow {x}_{j}=0$ , $j=1,\ldots ,k-1$ .
2. $\left\| \overrightarrow {x}\right\| =1$ .
Then $\lambda_{k}=Q(\overrightarrow {x}_{k})$ is the $k$ th eigenvalue of $A$ , $\lambda_{1} \geq \lambda_{2} \geq \cdots \geq \lambda_{k}$ and $\overrightarrow {x}_{k}$ is the corresponding eigenvectors of $A$ . Apply the maximum principle to the symmetric $2 \times 2$ matrix

A = \begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}.

Origin: Stated according to the original source

Problem 5.1.36

Let $A$ be an $n \times n$ matrix. An $n \times n$ matrix can have at most $n$ linearly independent eigenvectors. Now assume that $A$ has $n+1$ eigenvectors (at least one must be linearly dependent) such that any $n$ of them are linearly independent. Show that $A$ is a scalar multiple of the identity matrix $I_{n}$ .

Origin: Stated according to the original source

Problem 5.1.37

An $n \times n$ stochastic matrix $P$ satisfies the following conditions

p_{ij} \geq 0 \quad \text{for all } i,j=1,2,\ldots ,n

and

\sum _{i=1}^{n} p_{ij} = 1 \quad \text{for all } j=1,2,\ldots ,n.

Show that a stochastic matrix always has at least one eigenvalue equal to one.

Origin: Stated according to the original source

Problem 5.1.38

(i)(ii)(iii)(iv)

The matrix difference equation

\overrightarrow {p}(t+1) = M\overrightarrow {p}(t), \qquad t=0,1,2,\ldots

with the column vector (vector of probabilities)

\overrightarrow {p}(t) = (p_{1}(t), p_{2}(t), \ldots , p_{n}(t))^{T}

and the $n \times n$ matrix

M = \begin{pmatrix} (1-w) & 0.5w & 0 & & 0.5w \\ 0.5w & (1-w) & 0.5w & & 0 \\ 0 & 0.5w & (1-w) & & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0.5w & 0 & 0 & & (1-w) \end{pmatrix}

plays a role in random walk in one dimension. $M$ is called the transition probability matrix and $w$ denotes the probability $w \in [0,1]$ that at a given time step the particle jumps to either of its nearest neighbor sites and the probability that the particle does not jump either to the right of left is $(1-w)$ . The matrix $M$ is of the type known as circulant matrix. Such an $n \times n$ matrix is of the form

C = \begin{pmatrix} c_{0} & c_{1} & c_{2} & \cdots & c_{n-1} \\ c_{n-1} & c_{0} & c_{1} & \cdots & c_{n-2} \\ c_{n-2} & c_{n-1} & c_{0} & \cdots & c_{n-3} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ c_{1} & c_{2} & c_{3} & & c_{0} \end{pmatrix}

with the normalized eigenvectors

\overrightarrow {e}_{j} = \frac{1}{\sqrt{n}}\begin{pmatrix} 1 \\ e^{2\pi ij/n} \\ \vdots \\ e^{2(n-1)\pi ij/n} \end{pmatrix}, \qquad j=1,\ldots ,n.

Origin: Stated according to the original source

(i)

Use this result to find the eigenvalues of the matrix $C$ .

(ii)

Use (i) to find the eigenvalues of the matrix $M$ .

(iii)

Use (ii) to find $\overrightarrow {p}(t)$ ( $t=0,1,2,\ldots$ ), where we expand the initial distribution vector $\overrightarrow {p}(0)$ in terms of the eigenvectors

\overrightarrow {p}(0) = \sum _{k=1}^{n} a_{k}\overrightarrow {e}_{k} \quad \text{with} \quad \sum _{j=1}^{n} p_{j}(0) = 1.

(iv)

Assume that $\overrightarrow {p}(0) = \frac{1}{n}\begin{pmatrix} 1 & 1 & \cdots & 1 \end{pmatrix}^{T}$ . Give the time evolution of $\overrightarrow {p}(0)$ .

Problem 5.1.39

Let $n$ be a positive integer. Consider the $3 \times 3$ matrix with rows of elements summing to unity

M = \frac{1}{n}\begin{pmatrix} n-a-b & a & b \\ a & n-2a-c & a+c \\ c & a & n-a-c \end{pmatrix}

where the values of $a, b, c$ are such that $0 \leq a$ , $0 \leq b$ , $a+b \leq n$ , $2a+c \leq n$ . Thus the matrix is a stochastic matrix. Find the eigenvalues of $M$ .

Origin: Stated according to the original source

Problem 5.1.40

(i)(ii)

Let $U$ be a unitary matrix and $\overrightarrow {x}$ an eigenvector of $U$ with the corresponding eigenvalue $\lambda$ , i.e. $U\overrightarrow {x}=\lambda \overrightarrow {x}$ .

Origin: Stated according to the original source

(i)

Show that $U^{*}\overrightarrow {x}=\overline{\lambda }\overrightarrow {x}$ .

(ii)

Let $\lambda , \mu$ be distinct eigenvalues of a unitary matrix $U$ with the corresponding eigenvectors $\overrightarrow {x}$ and $\overrightarrow {y}$ , i.e. $U\overrightarrow {x}=\lambda \overrightarrow {x}$ , $U\overrightarrow {y}=\mu \overrightarrow {y}$ . Show that $\overrightarrow {x}^{*}\overrightarrow {y}=0$ .

Problem 5.1.41

Let $H, H_{0}, V$ be $n \times n$ matrices over $\mathbb {C}$ and $H=H_{0}+V$ . Let $z \in \mathbb {C}$ and assume that $z$ is chosen so that $(H_{0}-zI_{n})^{-1}$ and $(H-zI_{n})^{-1}$ exist. Show that

(H-zI_{n})^{-1} = (H_{0}-zI_{n})^{-1} - (H_{0}-zI_{n})^{-1}V(H-zI_{n})^{-1}.

This is called the second resolvent identity.

Origin: Stated according to the original source

Problem 5.1.42

An $n \times n$ matrix $A$ is called a Hadamard matrix if each entry of $A$ is $1$ or $-1$ and if the rows or columns of $A$ are orthogonal, i.e.

AA^{T} = nI_{n} \quad \text{or} \quad A^{T}A = nI_{n}.

Note that $AA^{T}=nI_{n}$ and $A^{T}A=nI_{n}$ are equivalent. Hadamard matrices $H_{n}$ of order $2^{n}$ can be generated recursively by defining

H_{1} = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, \qquad H_{n} = \begin{pmatrix} H_{n-1} & H_{n-1} \\ H_{n-1} & -H_{n-1} \end{pmatrix}

for $n \geq 2$ . Show that the eigenvalues of $H_{n}$ are given by $+2^{n/2}$ and $-2^{n/2}$ each of multiplicity $2^{n-1}$ .

Origin: Stated according to the original source

Problem 5.1.43

An $n \times n$ matrix $A$ over the complex numbers is called positive semidefinite (written as $A \geq 0$ ), if

\overrightarrow {x}^{*}A\overrightarrow {x} \geq 0 \quad \text{for all } \overrightarrow {x} \in \mathbb {C}^{n}.

Show that for every $A \geq 0$ , there exists a unique $B \geq 0$ so that $B^{2}=A$ . Thus $B$ is a square root of $A$ .

Origin: Stated according to the original source

Problem 5.1.44

(i)(ii)

Origin: Stated according to the original source

(i)

Consider the polynomial $p(x)=x^{2}-sx+d$ , $s,d \in \mathbb {C}$ . Find a $2 \times 2$ matrix $A$ such that its characteristic polynomial is $p$ .

(ii)

Consider the polynomial $q(x)=-x^{3}+sx^{2}-qx+d$ , $s,q,d \in \mathbb {C}$ . Find a $3 \times 3$ matrix $B$ such that its characteristic polynomial is $q$ .

Problem 5.1.45

Calculate the eigenvalues of the $4 \times 4$ matrix

A = \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & -1 & 0 \\ 1 & 0 & 0 & -1 \end{pmatrix}

by calculating the eigenvalues of $A^{2}$ .

Origin: Stated according to the original source

Problem 5.1.46

If $\left\{ A_{j}\right\}_{j=1}^{m}$ is a commuting family of matrices that is to say $A_{j}A_{k}=A_{k}A_{j}$ for every pair from the set, then there exists a unitary matrix $V$ such that for all $A_{j}$ in the set the matrix $\widetilde{A}_{j}=V^{*}A_{j}V$ is upper triangular. Apply this to the matrices

A_{1} = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}, \qquad A_{2} = \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}.

Origin: Stated according to the original source

Problem 5.1.47

(i)(ii)

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . The spectral radius of the matrix $A$ is the non-negative number defined by

\rho (A) := \max \left\{ \left|\lambda _{j}(A)\right| : 1 \leq j \leq n\right\}

where $\lambda_{j}(A)$ are the eigenvalues of $A$ . We define the norm of $A$ as

\left\| A\right\| := \sup _{\left\| \overrightarrow {x}\right\| =1} \left\| A\overrightarrow {x}\right\|

where $\left\| A\overrightarrow {x}\right\|$ denotes the Euclidean norm of the vector $A\overrightarrow {x}$ .

Origin: Stated according to the original source

(i)

Show that $\rho (A) \leq \left\| A\right\|$ .

(ii)

Consider the $2 \times 2$ matrix

A = \begin{pmatrix} 1/4 & 1/2 \\ 1/2 & 1/4 \end{pmatrix}.

Now $\rho (A) < 1$ . If $\rho (A) < 1$ , then $(I_{2}-A)^{-1} = I_{2}+A+A^{2}+\cdots$ . Calculate $(I_{2}-A)^{-1}$ . Calculate $(I_{2}-A)(I_{2}+A+A^{2}+\cdots +A^{k})$ .

Problem 5.1.48

Let $A$ be an $n \times n$ matrix with entries $a_{jk} \geq 0$ and with positive spectral radius $\rho$ . Then there is a (column) vector $\overrightarrow {x}$ with $x_{j} \geq 0$ and a (column) vector $\overrightarrow {y}$ such that the following conditions hold

A\overrightarrow {x} = \rho \overrightarrow {x}, \qquad \overrightarrow {y}^{T}A = \rho \overrightarrow {y}, \qquad \overrightarrow {y}^{T}\overrightarrow {x} = 1.

Consider the $2 \times 2$ symmetric matrix

B = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}.

Show that $B$ has a positive spectral radius. Find the vectors $\overrightarrow {x}$ and $\overrightarrow {y}$ .

Origin: Stated according to the original source

Problem 5.1.49

Consider a symmetric $2 \times 2$ matrix $A$ over $\mathbb {R}$ with $a_{11}>0$ , $a_{22}>0$ , $a_{12}<0$ and $a_{jj}>\left|a_{12}\right|$ for $j=1,2$ . Is the matrix $A$ positive definite?

Origin: Stated according to the original source

Problem 5.1.50

Let $A$ be a positive definite $n \times n$ matrix. Show that $A^{-1}$ exists and is also positive definite.

Origin: Stated according to the original source

Problem 5.1.51

Find the eigenvalues of the symmetric matrices

A_{3} = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix}, \qquad A_{4} = \begin{pmatrix} 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{pmatrix}, \qquad A_{5} = \begin{pmatrix} 0 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 \end{pmatrix}.

Origin: Stated according to the original source

Problem 5.1.52

Consider the $n \times n$ cyclic matrix

A = \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} & \cdots & a_{1,n-1} & a_{1n} \\ a_{1n} & a_{11} & a_{12} & a_{13} & \cdots & a_{1,n-2} & a_{1,n-1} \\ a_{1,n-1} & a_{1n} & a_{11} & a_{12} & \cdots & a_{n-3} & a_{n-2} \\ \vdots & \vdots & \vdots & \vdots & & \ddots & \vdots \\ a_{12} & a_{13} & a_{14} & a_{15} & \cdots & a_{1n} & a_{11} \end{pmatrix}

where $a_{jk} \in \mathbb {R}$ . Show that

\overrightarrow {v} = \frac{1}{\sqrt{n}}\begin{pmatrix} \epsilon ^{2k} \\ \epsilon ^{4k} \\ \vdots \\ \epsilon ^{2(n-1)k} \\ 1 \end{pmatrix}, \qquad \epsilon \equiv e^{i\pi /n}, \qquad 1 \leq k \leq n

is a normalized eigenvector of $A$ . Find the eigenvalues.

Origin: Stated according to the original source

Problem 5.1.53

(i)(ii)

Origin: Stated according to the original source

(i)

Let $a,b \in \mathbb {R}$ . Find on inspection two eigenvectors and the corresponding eigenvalues of the symmetric $4 \times 4$ matrix

\begin{pmatrix} a & 0 & 0 & b \\ 0 & a & 0 & b \\ 0 & 0 & a & b \\ b & b & b & 0 \end{pmatrix}.

(ii)

Let $a,b \in \mathbb {R}$ . Find on inspection two eigenvectors and the corresponding eigenvalues of the $4 \times 4$ matrix

\begin{pmatrix} a & 0 & 0 & b \\ 0 & a & 0 & b \\ 0 & 0 & -a & b \\ b & b & b & 0 \end{pmatrix}.

Problem 5.1.54

Consider the two $4 \times 4$ permutation matrices

S = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}, \qquad T = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}.

Show that the two matrices have the same (normalized) eigenvectors. Find the commutator $\left[S,T\right]$ .

Origin: Stated according to the original source

Problem 5.1.55

(i)(ii)

Let $\alpha \in \mathbb {R}$ . Consider the symmetric $4 \times 4$ matrix

A(\alpha ) = \begin{pmatrix} 1 & \alpha & 0 & 0 \\ \alpha & 2 & 2\alpha & 0 \\ 0 & 2\alpha & 3 & \alpha \\ 0 & 0 & \alpha & 4 \end{pmatrix}.

Origin: Stated according to the original source

(i)

Find the characteristic equation.

(ii)

Show that

\begin{align} \lambda _{1}+\lambda _{2}+\lambda _{3}+\lambda _{4} & = 10 \\ \lambda _{1}\lambda _{2}+\lambda _{1}\lambda _{3}+\lambda _{1}\lambda _{4}+\lambda _{2}\lambda _{3}+\lambda _{2}\lambda _{4}+\lambda _{3}\lambda _{4} & = 35-6\alpha ^{2} \\ \lambda _{1}\lambda _{2}\lambda _{3}+\lambda _{1}\lambda _{2}\lambda _{4}+\lambda _{1}\lambda _{3}\lambda _{4}+\lambda _{2}\lambda _{3}\lambda _{4} & = 50-30\alpha ^{2} \\ \lambda _{1}\lambda _{2}\lambda _{3}\lambda _{4} & = 24-30\alpha ^{2}+\alpha ^{4} \end{align}

where $\lambda_{1}, \lambda_{2}, \lambda_{3}, \lambda_{4}$ denote the eigenvalues.

Problem 5.1.56

Let $B$ be a $2 \times 2$ matrix with eigenvalues $\lambda_{1}$ and $\lambda_{2}$ . Find the eigenvalues of the $4 \times 4$ matrix

X = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ b_{11} & b_{12} & 0 & 0 \\ b_{21} & b_{22} & 0 & 0 \end{pmatrix}.

Let $\overrightarrow {v}$ be an eigenvector of $B$ with eigenvalue $\lambda$ . What can be said about an eigenvector of the $4 \times 4$ matrix $X$ given by eigenvector $\overrightarrow {v}$ and eigenvalue of $B$ ?

Origin: Stated according to the original source

Problem 5.1.57

Let $A, B$ be two $n \times n$ matrices over $\mathbb {C}$ . The set of all matrices of the form $A-\lambda B$ with $\lambda \in \mathbb {C}$ is said to be a pencil. The eigenvalues of the pencil are elements of the set $\lambda (A,B)$ defined by

\lambda (A,B) := \left\{ z \in \mathbb {C} : \operatorname {det}\left(A-zB\right)=0\right\} .

If $\lambda \in \lambda (A,B)$ and $A\overrightarrow {v}=\lambda B\overrightarrow {v}$ , $\overrightarrow {v} \neq \overrightarrow {0}$ then $\overrightarrow {v}$ is referred to as an eigenvector of $A-\lambda B$ . Note that $\lambda$ may be finite, empty or infinite. Let

A = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}, \qquad B = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & -1 & 0 \\ 1 & 0 & 0 & -1 \end{pmatrix}.

Find the eigenvalue of the pencil.

Origin: Stated according to the original source

Problem 5.1.58

Let $A$ be an $n \times n$ matrix with eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$ . Let $c \in \mathbb {C} \setminus \left\{ 0\right\}$ . What are the eigenvalues of $cA$ ?

Origin: Stated according to the original source

Problem 5.1.59

Let $A, B$ be $n \times n$ matrices over $\mathbb {C}$ . Let $\alpha , \beta \in \mathbb {C}$ . Assume that $A^{2}=I_{n}$ and $B^{2}=I_{n}$ and $AB+BA=0_{n}$ . What can be said about the eigenvalues of $\alpha A+\beta B$ ?

Origin: Stated according to the original source

Problem 5.1.60

(i)(ii)

Origin: Stated according to the original source

(i)

Let $A, B$ be $n \times n$ matrices. Show that $AB$ and $BA$ have the same eigenvalues.

(ii)

Can we conclude that every eigenvector of $AB$ is also an eigenvector of $BA$ ?

Problem 5.1.61

(i)(ii)

Origin: Stated according to the original source

(i)

Show that if $A$ is an $n \times m$ matrix and if $B$ is an $m \times n$ matrix, then $\lambda \neq 0$ is an eigenvalue of the $n \times n$ matrix $AB$ if and only if $\lambda$ is an eigenvalue of the $m \times m$ matrix $BA$ . Show that if $m=n$ then the conclusion is true even for $\lambda =0$ .

(ii)

Let $M$ be an $m \times n$ matrix ( $m<n$ ) over $\mathbb {R}$ . Show that at least one eigenvalue of the $n \times n$ matrix $M^{T}M$ is equal to $0$ . Show that the eigenvalues of the $m \times m$ matrix $MM^{T}$ are also eigenvalues of $M^{T}M$ .

Problem 5.1.62

We know that a hermitian matrix has only real eigenvalues. Can we conclude that a matrix with only real eigenvalues is hermitian?

Origin: Stated according to the original source

Problem 5.1.63

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . Show that the eigenvalues of $A^{*}A$ are nonnegative.

Origin: Stated according to the original source

Problem 5.1.64

Let $n$ be a positive integer. Consider the $2 \times 2$ matrix

T_{n} = \begin{pmatrix} 2n & 4n^{2}-1 \\ 1 & 2n \end{pmatrix} \quad \Rightarrow \quad \operatorname {det}\left(T_{n}\right) = 1.

Show that the eigenvalues of $T_{n}$ are real and not of absolute value $1$ .

Origin: Stated according to the original source

Problem 5.1.65

Let $L_{n}$ be the $n \times n$ matrix

L_{n} = \begin{pmatrix} n-1 & -1 & -1 & & -1 \\ -1 & n-1 & -1 & & -1 \\ \vdots & \vdots & \ddots & & \vdots \\ -1 & -1 & -1 & \ddots & -1 \\ -1 & -1 & -1 & \cdots & n-1 \end{pmatrix}.

Find the eigenvalues.

Origin: Stated according to the original source

Problem 5.1.66

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . Assume that $A^{2}=-I_{n}$ . What can be said about the eigenvalues of $A$ ?

Origin: Stated according to the original source

Problem 5.1.67

(i)(ii)(iii)

Consider the symmetric $6 \times 6$ matrix over $\mathbb {R}$

A = \begin{pmatrix} 0 & 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & -1 & -1 & 1 \\ 1 & 1 & 0 & 1 & -1 & -1 \\ 1 & -1 & 1 & 0 & 1 & -1 \\ 1 & -1 & -1 & 1 & 0 & 1 \\ 1 & 1 & -1 & -1 & 1 & 0 \end{pmatrix}.

This matrix plays a role in the construction of the icosahedron which is a regular polyhedron with $20$ identical equilateral triangular faces, $30$ edges and $12$ vertices.

Origin: Stated according to the original source

(i)

Find the eigenvalues of this matrix.

(ii)

Consider the matrix $A+\sqrt{5}I_{6}$ . Find the eigenvalues.

(iii)

The matrix $A+\sqrt{5}I_{6}$ induces an Euclidean structure on the quotient space $\mathbb {R}^{6}/\ker (A+\sqrt{5}I_{6})$ . Find the dimension of $\ker (A+\sqrt{5}I_{6})$ .

Problem 5.1.68

Let $A$ be an $n \times n$ matrix over $\mathbb {C}^{n}$ . Let $\lambda$ be an eigenvalue of $A$ . A generalized eigenvector $\overrightarrow {x} \in \mathbb {C}^{n}$ of $A$ corresponding to the eigenvalue $\lambda$ is a nontrivial solution of

(A-\lambda I_{n})^{j}\overrightarrow {x} = \overrightarrow {0}_{n}

for some $j \in \left\{ 1,2,\ldots \right\}$ , where $\overrightarrow {0}_{n}$ is the $n$ -dimensional zero vector. For $j=1$ we find the eigenvectors. It follows that $\overrightarrow {x}$ is a generalized eigenvector of $A$ corresponding to $\lambda$ if and only if

(A-\lambda I_{n})^{n}\overrightarrow {x} = \overrightarrow {0}_{n}.

Find the eigenvectors and generalized eigenvectors of the nonnormal matrix

\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & -1 & 0 \end{pmatrix}.

Origin: Stated according to the original source

Problem 5.1.69

(i)(ii)(iii)(iv)

Consider the $2n \times 2n$ matrix

J := \begin{pmatrix} 0_{n} & I_{n} \\ -I_{n} & 0_{n} \end{pmatrix} \in \mathbb {R}^{2n \times 2n}.

A matrix $S \in \mathbb {R}^{2n \times 2n}$ is called a symplectic matrix if $S^{T}JS=J$ .

Origin: Stated according to the original source

(i)

Show that symplectic matrices are nonsingular.

(ii)

Show that the product of two symplectic matrices $S_{1}$ and $S_{2}$ is also symplectic.

(iii)

Show that if $S$ is symplectic $S^{-1}$ and $S^{T}$ are also symplectic.

(iv)

Let $S$ be a symplectic matrix. Show that if $\lambda \in \sigma (S)$ , then $\lambda^{-1} \in \sigma (S)$ , where $\sigma (S)$ denotes the spectrum of $S$ .

Problem 5.1.70

Let $A, B$ be $n \times n$ matrices over $\mathbb {C}$ and $\overrightarrow {u}$ a nonzero vector in $\mathbb {C}^{n}$ . Assume that $\left[A,B\right]=A$ and $A\overrightarrow {v}=\lambda \overrightarrow {v}$ . Find $(AB)\overrightarrow {v}$ .

Origin: Stated according to the original source

Problem 5.1.71

Let $A, B$ be $n \times n$ matrices over $c \in \mathbb {C}$ with $\left[A,B\right]=0_{n}$ . Then $\left[A+cI_{n}, B+cI_{n}\right]=0_{n}$ , where $c \in \mathbb {C}$ . Let $\overrightarrow {v}$ be an eigenvector of the $n \times n$ matrix $A$ with eigenvalue $\lambda$ . Show that $\overrightarrow {v}$ is also an eigenvector of $A+cI_{n}$ , where $c \in \mathbb {C}$ .

Origin: Stated according to the original source

Problem 5.1.72

Let $A$ be an $n \times n$ normal matrix over $\mathbb {C}$ . How would one apply genetic algorithms to find the eigenvalues of $A$ . This means we have to construct a fitness function $f$ with the minima as the eigenvalues. The eigenvalue equation is given by $A\overrightarrow {x}=z\overrightarrow {x}$ ( $z \in \mathbb {C}$ and $\overrightarrow {x} \in \mathbb {C}^{n}$ with $\overrightarrow {x} \neq \overrightarrow {0}$ ). The characteristic equation is $p(z) \equiv \operatorname {det}\left(A-zI_{n}\right)=0$ . What would be a fitness function? Apply it to the matrices

B = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad C = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}, \qquad D = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}.

Origin: Stated according to the original source

Problem 5.1.73

Let $\overrightarrow {v}$ be a nonzero column vector in $\mathbb {R}^{n}$ . Matrix multiplication is associative. Then we have $(\overrightarrow {v}\overrightarrow {v}^{T})\overrightarrow {v} = \overrightarrow {v}(\overrightarrow {v}^{T}\overrightarrow {v})$ . Discuss.

Origin: Stated according to the original source

Problem 5.1.74

Let $n \geq 2$ and even. Consider an $n \times n$ hermitian matrix $A$ . Thus the eigenvalues are real. Assume we have the information that if $\lambda$ is an eigenvalue then $-\lambda$ is also an eigenvalue of $A$ . How can the calculation of the eigenvalues be simplified with this information?

Origin: Stated according to the original source

Problem 5.1.75

Let $A, B$ be two nonzero $n \times n$ matrices over $\mathbb {C}$ . Let $A\overrightarrow {v}=\lambda \overrightarrow {v}$ be the eigenvalue equation for $A$ . Assume that $\left[A,B\right]=0_{n}$ . Then from $\left[A,B\right]\overrightarrow {v}=\overrightarrow {0}$ it follows that

\left[A,B\right]\overrightarrow {v} = (AB-BA)\overrightarrow {v} = A(B\overrightarrow {v})-B(A\overrightarrow {v}) = \overrightarrow {0}.

Therefore $A(B\overrightarrow {v})=\lambda (B\overrightarrow {v})$ . If $B\overrightarrow {v} \neq \overrightarrow {0}$ we find that $B\overrightarrow {v}$ is an eigenvector of $A$ with eigenvalue $\lambda$ . Apply it to $A=\sigma_{1} \otimes \sigma_{2}$ and $B=\sigma_{3} \otimes \sigma_{3}$ .

Origin: Stated according to the original source

Problem 5.1.76

Let $A, B$ be two nonzero $n \times n$ matrices over $\mathbb {C}$ . Let $A\overrightarrow {v}=\lambda \overrightarrow {v}$ be the eigenvalue equation for $A$ . Assume that $\left[A,B\right]_{+}=0_{n}$ . Then from $\left[A,B\right]_{+}\overrightarrow {v}=\overrightarrow {0}$ it follows that

\left[A,B\right]_{+}\overrightarrow {v} = (AB+BA)\overrightarrow {v} = A(B\overrightarrow {v})+B(A\overrightarrow {v}) = \overrightarrow {0}.

Therefore $A(B\overrightarrow {v})=-\lambda (B\overrightarrow {v})$ . If $B\overrightarrow {v} \neq \overrightarrow {0}$ we have an eigenvalue equation with eigenvalue $-\lambda$ . Apply it to $A=\sigma_{1}$ and $B=\sigma_{2}$ , where $\left[\sigma_{1},\sigma_{2}\right]_{+}=0_{2}$ .

Origin: Stated according to the original source

Problem 5.1.77

(i)(ii)(iii)

Consider the real symmetric $3 \times 3$ matrices

A = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix}.

Origin: Stated according to the original source

(i)

Find the eigenvalues and normalized eigenvectors for the matrices $A$ and $B$ .

(ii)

Find the commutator $\left[A,B\right]$ .

(iii)

Discuss the results from (ii) and (i) with respect to the eigenvectors.

Problem 5.1.78

Consider the Pauli spin matrix $\sigma_{2}$ with the eigenvalues $+1, -1$ and the corresponding normalized eigenvectors

\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ i \end{pmatrix}, \qquad \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -i \end{pmatrix}.

Then $\sigma_{2} \otimes \sigma_{2}$ admits the eigenvalues $+1$ (twice) and $-1$ (twice). Show that there are product states as eigenvectors and two sets of entangled eigenvectors for $\sigma_{2} \otimes \sigma_{2}$ .

Origin: Stated according to the original source

Problem 5.1.79

(i)(ii)

Origin: Stated according to the original source

(i)

Let $A$ be an invertible $n \times n$ matrix. Given the eigenvalues and eigenvectors of $A$ . What can be said about the eigenvalues of $A \otimes A^{-1}+A^{-1} \otimes A$ ?

(ii)

Let $c \in \mathbb {R}$ and $A$ be a normal $n \times n$ matrix. Assume that $A$ is invertible. The eigenvalue equation is given by $A\overrightarrow {v}_{j}=\lambda_{j}\overrightarrow {v}_{j}$ for $j=1,\ldots ,n$ and $\lambda_{j} \neq 0$ for all $j=1,\ldots ,n$ . Find all the eigenvalues and eigenvectors of the matrix

A \otimes A^{-1} + c(A^{-1} \otimes A).

Find $\operatorname {det}\left(A \otimes A^{-1}\right)$ and $\operatorname {tr}(A \otimes A^{-1})$ .

Problem 5.1.80

Let $M$ be an $n \times n$ matrix over $\mathbb {R}$ given by

M = D + \overrightarrow {u} \otimes \overrightarrow {v}^{T}

where $D$ is a diagonal matrix over $\mathbb {R}$ and $\overrightarrow {u}, \overrightarrow {v}$ are column vectors in $\mathbb {R}^{n}$ . The eigenvalue equation is given by

M\overrightarrow {x} \equiv (D+\overrightarrow {u} \otimes \overrightarrow {v}^{T})\overrightarrow {x} = \lambda \overrightarrow {x}.

Assume that $\sum_{j=1}^{n} u_{j}v_{j} \neq 0$ . Show that

\sum _{k=1}^{n} \frac{u_{k}v_{k}}{(d_{k}-\lambda )} + 1 = 0

where $d_{k}$ are the diagonal elements of $D$ .

Origin: Stated according to the original source

Problem 5.1.81

Let $A$ be an $n \times n$ matrix over $\mathbb {R}$ with eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$ . Find the eigenvalues of the $2n \times 2n$ matrix

\begin{pmatrix} 0_{n} & A \\ -A & 0_{n} \end{pmatrix} \equiv \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \otimes A.

Is this matrix skew-symmetric over $\mathbb {R}$ ?

Origin: Stated according to the original source

Problem 5.1.82

Consider the nonnormal $2 \times 2$ matrix

A = \begin{pmatrix} 3 & 2 \\ 4 & 3 \end{pmatrix}.

For $j \geq 1$ , let $d_{j}$ be the greatest common divisor of the entries of $A^{j}-I_{2}$ . Show that

\lim _{j \rightarrow \infty } d_{j} = \infty .

Use the eigenvalues of $A$ and the characteristic polynomial.

Origin: Stated according to the original source

Problem 5.1.83

(i)(ii)(iii)

The $2 \times 3$ matrix

A = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix}

is the starting matrix in the construction of the Hironaka curve utilizing the Kronecker product.

Origin: Stated according to the original source

(i)

Find $A^{T}$ and the rank of $A$ and $A^{T}$ .

(ii)

Find $AA^{T}$ and the eigenvalues of $AA^{T}$ . Find $A^{T}A$ and the eigenvalues of $A^{T}A$ . Compare the eigenvalues of $AA^{T}$ and $A^{T}A$ . Discuss.

(iii)

Calculate $A \otimes A$ and $A^{T} \otimes A^{T}$ . Calculate the eigenvalues of

(A \otimes A)(A^{T} \otimes A^{T}), \qquad (A^{T} \otimes A^{T})(A \otimes A).

Problem 5.1.84

(i)(ii)

Consider the unitary $4 \times 4$ matrix

U = \begin{pmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \otimes \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} = -i\sigma _{2} \otimes I_{2}.

Origin: Stated according to the original source

(i)

Find the eigenvalues and normalized eigenvectors of $U$ .

(ii)

Can the normalized eigenvectors of $U$ be written as Kronecker product of two normalized vectors in $\mathbb {C}^{2}$ . If so they are not entangled.

Problem 5.1.85

Consider the Hilbert space $\mathbb {C}^{n}$ . Let $A, B, C$ be $n \times n$ matrices acting in $\mathbb {C}^{n}$ . We consider the nonlinear eigenvalue problem

A\overrightarrow {v} = \lambda B\overrightarrow {v} + \lambda ^{2}C\overrightarrow {v}

where $\overrightarrow {v} \in \mathbb {C}^{n}$ and $\overrightarrow {v} \neq \overrightarrow {0}$ . Let $\sigma_{1}, \sigma_{2}, \sigma_{3}$ be the Pauli spin matrices. Find the solutions of the nonlinear eigenvalue problem

\sigma _{1}\overrightarrow {v} = \lambda \sigma _{2}\overrightarrow {v} + \lambda ^{2}\sigma _{3}\overrightarrow {v}

where $\overrightarrow {v} \in \mathbb {C}^{2}$ and $\overrightarrow {v} \neq \overrightarrow {0}$ .

Origin: Stated according to the original source

Problem 5.1.86

Let $A$ be an $n \times n$ symmetric matrix over $\mathbb {R}$ . Since $A$ is symmetric over $\mathbb {R}$ there exists a set of orthonormal eigenvectors $\overrightarrow {v}_{1}, \overrightarrow {v}_{2}, \ldots , \overrightarrow {v}_{n}$ which form an orthonormal basis in $\mathbb {R}^{n}$ . Let $\overrightarrow {x} \in \mathbb {R}^{n}$ be a reasonably good approximation to an eigenvector, say $\overrightarrow {v}_{1}$ . Calculate

R := \frac{\overrightarrow {x}^{T}A\overrightarrow {x}}{\overrightarrow {x}^{T}\overrightarrow {x}}.

The quotient is called Rayleigh quotient. Discuss.

Origin: Stated according to the original source

Section 5.2Supplementary Problems[71/0%]

Problem 5.2.1

(i)(ii)(iii)(iv)(v)(vi)(vii)

Origin: Stated according to the original source

(i)

Let $\theta \in \left[0,\pi /2\right]$ . Find the trace, determinant, eigenvalues and eigenvectors of the $2 \times 2$ matrices

M_{1} = \frac{1}{2}\begin{pmatrix} \tan ^{2}(\theta ) & -\tan (\theta ) \\ -\tan (\theta ) & 1 \end{pmatrix}, \qquad M_{2} = \frac{1}{2}\begin{pmatrix} \tan ^{2}(\theta ) & \tan (\theta ) \\ \tan (\theta ) & 1 \end{pmatrix}.

(ii)

Find the eigenvalues and normalized eigenvectors of the matrix ( $\phi \in \left[0,2\pi \right]$ )

A(\phi ) = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & e^{i\phi } \\ 1 & e^{-i\phi } \end{pmatrix}.

Is the matrix invertible? Make the decision by looking at the eigenvalues. If so find the inverse matrix.

(iii)

Let $\epsilon \in \mathbb {R}$ . Find the eigenvalues and eigenvectors of

A(\epsilon ) = \frac{1}{\sqrt{1+\epsilon ^{2}}}\begin{pmatrix} \epsilon & 1 \\ 1 & \epsilon \end{pmatrix}.

For which $\epsilon$ is $A(\epsilon )$ not invertible? Let $\epsilon \in \left[0,1\right]$ . Consider the $2 \times 2$ matrix

B(\epsilon ) = \frac{1}{\sqrt{1+\epsilon ^{2}}}\begin{pmatrix} 1 & \epsilon \\ \epsilon & -1 \end{pmatrix}.

For $\epsilon =0$ we have the Pauli spin matrix $\sigma_{3}$ and for $\epsilon =1$ we have the Hadamard matrix. Find the eigenvalues and eigenvectors of $A(\epsilon )$ .

(iv)

Let $\epsilon \in \mathbb {R}$ . Find the eigenvalues $\lambda_{+}(\epsilon ), \lambda_{-}(\epsilon )$ and normalized eigenvectors of the matrix

A(\epsilon ) = \begin{pmatrix} \sinh (\epsilon ) & 1 \\ 1 & -\sinh (\epsilon ) \end{pmatrix}.

Find the shortest distance between $\lambda_{+}(\epsilon )$ and $\lambda_{-}(\epsilon )$ .

(v)

Let $\theta \in \mathbb {R}$ . Show that the eigenvalues of the hermitian matrix

A(\theta ) = \begin{pmatrix} 1 & \cos (\theta ) \\ \cos (\theta ) & 1 \end{pmatrix}

are given by $\lambda_{\pm }=1\pm \cos (\theta )$ and thus cannot be negative.

(vi)

Let $\tau := (1+\sqrt{5})/2$ be the golden ratio. Consider the modular $2 \times 2$ matrix

M = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}.

Show that the eigenvalues are given by $\tau$ and $-\tau^{-1}$ , where $\tau^{-1}=(\sqrt{5}-1)/2$ and the normalized eigenvectors are

\overrightarrow {v}_{1} = \frac{1}{1+\tau ^{2}}\begin{pmatrix} \tau \\ 1 \end{pmatrix}, \qquad \overrightarrow {v}_{2} = \frac{1}{1+\tau ^{2}}\begin{pmatrix} -\tau ^{-1} \\ 1 \end{pmatrix}.

(vii)

Consider the two $2 \times 2$ matrices

A_{\pm } = \begin{pmatrix} -1/2 & \mp 3/2 \\ \pm 1/2 & -1/2 \end{pmatrix}.

Show that $\operatorname {det}\left(A_{\pm }\right)=1$ , $\operatorname {tr}(A_{\pm })=-1$ , $A_{\pm }^{-1}=A_{\mp }$ , $A_{\pm }^{3}=I_{2}$ and that the eigenvalues of the two matrices are the same, namely

\frac{1}{2}(\sqrt{3}i-1), \qquad -\frac{1}{2}(\sqrt{3}i+1).

Problem 5.2.2

(i)(ii)

Origin: Stated according to the original source

(i)

The symmetric $3 \times 3$ matrix over $\mathbb {R}$

A = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix}

plays a role for the chemical compounds $ZnS$ and $NaCl$ . Find the eigenvalues and eigenvectors of $A$ . Then find the inverse of $A$ . Find all $\overrightarrow {v}$ such that $A\overrightarrow {v}=\overrightarrow {v}$ .

(ii)

Let $a,b,c \in \mathbb {R}$ . Find the eigenvalues and eigenvectors of the symmetric matrix

\begin{pmatrix} 0 & a & b \\ a & 0 & c \\ b & c & 0 \end{pmatrix}.

The characteristic equation is $\lambda^{3}-(a^{2}+b^{2}+c^{2})\lambda -2abc=0$ .

Problem 5.2.3

(i)(ii)

Origin: Stated according to the original source

(i)

Find the eigenvalues and eigenvectors of the orthogonal matrices

R(\alpha ) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos (\alpha ) & \sin (\alpha ) \\ 0 & -\sin (\alpha ) & \cos (\alpha ) \end{pmatrix}, \qquad S(\beta ) = \begin{pmatrix} \cos (\beta ) & \sin (\beta ) & 0 \\ -\sin (\beta ) & \cos (\beta ) & 0 \\ 0 & 0 & 1 \end{pmatrix}.

(ii)

Find the eigenvalues and eigenvectors of $R(\alpha )S(\beta )$ .

Problem 5.2.4

(i)(ii)(iii)(iv)(v)

Origin: Stated according to the original source

(i)

Find the eigenvalues and eigenvectors of the $3 \times 3$ matrices

\begin{pmatrix} 0 & 0 & a_{13} \\ 1 & 0 & a_{23} \\ 0 & 1 & a_{33} \end{pmatrix}, \qquad \begin{pmatrix} 0 & a_{12} & 0 \\ a_{21} & 0 & a_{23} \\ 0 & a_{32} & a_{33} \end{pmatrix}.

(ii)

Let $A$ be a normal $2 \times 2$ matrix over $\mathbb {C}$ with eigenvalues $\lambda_{1}, \lambda_{2}$ and corresponding eigenvectors $\overrightarrow {v}_{1}, \overrightarrow {v}_{2}$ , respectively. Let $c \in \mathbb {C}$ . What can be said about the eigenvalues and eigenvectors of the $3 \times 3$ matrices

B_{1} = \begin{pmatrix} c & 0 & 0 \\ 0 & a_{11} & a_{12} \\ 0 & a_{21} & a_{22} \end{pmatrix}, \qquad B_{2} = \begin{pmatrix} a_{11} & 0 & a_{12} \\ 0 & c & 0 \\ a_{21} & 0 & a_{22} \end{pmatrix}, \qquad B_{3} = \begin{pmatrix} a_{11} & a_{12} & 0 \\ a_{21} & a_{22} & 0 \\ 0 & 0 & c \end{pmatrix}?

(iii)

Let $\phi \in \mathbb {R}$ . Find the eigenvalues and eigenvectors of the $3 \times 3$ matrices

X(\phi ) = \begin{pmatrix} 0 & 0 & e^{i\phi } \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}, \qquad Y(\phi ) = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ e^{i\phi } & 0 & 0 \end{pmatrix}.

(iv)

Consider the $3 \times 3$ matrix

A = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 1 \end{pmatrix}.

Show that the characteristic polynomial is given by $p(\lambda )=\lambda^{3}-\lambda^{2}-1$ . Find the eigenvalues. Calculate $A \otimes A$ , $A \otimes A \otimes A$ , $A \otimes A \otimes A \otimes A$ . Then identify $1$ with a black square and $0$ with a white square. Draw the pictures.

(v)

Let $m>0$ and $\theta \in \mathbb {R}$ . Consider the three $3 \times 3$ matrices

M_{1}(\theta ) = m\begin{pmatrix} 0 & \sin (\theta ) & 0 \\ \sin (\theta ) & 0 & \cos (\theta ) \\ 0 & \cos (\theta ) & 0 \end{pmatrix}, \qquad M_{2}(\theta ) = m\begin{pmatrix} 0 & 0 & \sin (\theta ) \\ 0 & 0 & \cos (\theta ) \\ \sin (\theta ) & \cos (\theta ) & 0 \end{pmatrix},

M_{3}(\theta ) = m\begin{pmatrix} 0 & \sin (\theta ) & \cos (\theta ) \\ \sin (\theta ) & 0 & 0 \\ \cos (\theta ) & 0 & 0 \end{pmatrix}.

Find the eigenvalues and eigenvectors of the matrices. These matrices play a role for the Majorana neutrino.

Problem 5.2.5

(i)(ii)

Let $\epsilon \in \mathbb {R}$ .

Origin: Stated according to the original source

(i)

Find the eigenvalues and eigenvectors of the matrices

\begin{pmatrix} 1 & \epsilon \\ 1 & 1 \end{pmatrix}, \qquad \begin{pmatrix} 1 & 0 & \epsilon \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{pmatrix}, \qquad \begin{pmatrix} 1 & 0 & 0 & \epsilon \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \end{pmatrix}.

Extend to $n \times n$ matrices.

(ii)

Find the eigenvalues and eigenvectors of the matrices

\begin{pmatrix} 0 & 1 \\ \epsilon & 0 \end{pmatrix}, \qquad \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ \epsilon & 0 & 0 \end{pmatrix}, \qquad \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \epsilon & 0 & 0 & 0 \end{pmatrix}.

Extend to the $n \times n$ case.

Problem 5.2.6

(i)(ii)

Origin: Stated according to the original source

(i)

Consider the $5 \times 4$ matrix

A = \begin{pmatrix} 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 \end{pmatrix}.

Find the eigenvalues of the matrices $A^{T}A$ and $AA^{T}$ by only calculating the eigenvalues of $A^{T}A$ .

(ii)

Let say we want to calculate the eigenvalues of the $4 \times 4$ matrix

B = \begin{pmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{pmatrix}.

How could we utilize the facts that

\begin{pmatrix} 0 & 1 \\ 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{pmatrix} = B, \qquad \begin{pmatrix} 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}

and $\operatorname {tr}(B)=4$ and thus avoid solving $\operatorname {det}\left(B-\lambda I_{4}\right)=0$ ?

Problem 5.2.7

Let $A$ be an $n \times n$ diagonalizable matrix with distinct eigenvalues $\lambda_{j}$ ( $j=1,\ldots ,n$ ). Show that the Vandermonde matrix

V = \begin{pmatrix} 1 & 1 & & 1 \\ \lambda _{1} & \lambda _{2} & & \lambda _{n} \\ \lambda _{1}^{2} & \lambda _{2}^{2} & & \lambda _{n}^{2} \\ \vdots & \vdots & & \vdots \\ \lambda _{1}^{n-1} & \lambda _{2}^{n-1} & & \lambda _{n}^{n-1} \end{pmatrix}

is nonsingular.

Origin: Stated according to the original source

Problem 5.2.8

(i)(ii)

Origin: Stated according to the original source

(i)

Let $\sigma_{1}, \sigma_{2}, \sigma_{3}$ be the Pauli spin matrices. Solve the nonlinear eigenvalue problem

\sigma _{1}\overrightarrow {u} = \lambda \sigma _{2}\overrightarrow {u} + \lambda ^{2}\sigma _{3}\overrightarrow {u}

where $\overrightarrow {u} \in \mathbb {C}^{2}$ and $\overrightarrow {u} \neq \overrightarrow {0}$ .

(ii)

Solve the nonlinear eigenvalue problem

(\sigma _{1} \otimes \sigma _{1})\overrightarrow {v} = \lambda (\sigma _{2} \otimes \sigma _{2})\overrightarrow {v} + \lambda ^{2}(\sigma _{3} \otimes \sigma _{3})\overrightarrow {v}

where $\overrightarrow {v} \in \mathbb {C}^{4}$ and $\overrightarrow {v} \neq \overrightarrow {0}$ . Compare the result to (i). Discuss.

Problem 5.2.9

Let $A, B, C, D, E, F, G, H$ be $2 \times 2$ matrices over $\mathbb {C}$ . We define the star product

\begin{pmatrix} A & B \\ C & D \end{pmatrix} \star \begin{pmatrix} E & F \\ G & H \end{pmatrix} := \begin{pmatrix} E & 0_{2} & 0_{2} & F \\ 0_{2} & A & B & 0_{2} \\ 0_{2} & C & D & 0_{2} \\ G & 0_{2} & 0_{2} & H \end{pmatrix}.

Thus the right-hand side is an $8 \times 8$ matrix. Assume we know the eigenvalues and eigenvectors of the two $4 \times 4$ matrices on the left-hand side. What can be said about the eigenvalues and eigenvectors of the $8 \times 8$ matrix of the right-hand side?

Origin: Stated according to the original source

Problem 5.2.10

Consider a $2 \times 2$ matrix $A=(a_{jk})$ over $\mathbb {R}$ with

\sum _{j,k=1}^{2} a_{jk} = 1, \qquad \operatorname {tr}(A) = 0.

What can be said about the eigenvalues of such a matrix?

Origin: Stated according to the original source

Problem 5.2.11

Consider the Hadamard matrix

U = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \equiv \frac{1}{\sqrt{2}}(\sigma _{3}+\sigma _{1}).

The eigenvalues of the Hadamard matrix are given by $+1$ and $-1$ with the corresponding normalized eigenvectors

\frac{1}{\sqrt{8}}\begin{pmatrix} \sqrt{4+2\sqrt{2}} \\ \sqrt{4-2\sqrt{2}} \end{pmatrix}, \qquad \frac{1}{\sqrt{8}}\begin{pmatrix} \sqrt{4-2\sqrt{2}} \\ -\sqrt{4+2\sqrt{2}} \end{pmatrix}.

How can this information be used to find the eigenvalues and eigenvectors of the Bell matrix

B = U \star U \equiv \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & -1 & 0 \\ 1 & 0 & 0 & -1 \end{pmatrix}?

Origin: Stated according to the original source

Problem 5.2.12

(i)(ii)

Origin: Stated according to the original source

(i)

Let $U$ be an $n \times n$ unitary matrix. Assume that $U^{*}=-U$ , i.e. the matrix is also skew-hermitian. Find the eigenvalues of such a matrix.

(ii)

Let $V$ be an $n \times n$ unitary matrix. What can be concluded about the eigenvalues of $V$ if $V^{*}=V^{T}$ ?

Problem 5.2.13

Find the eigenvalues of the staircase matrices

\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}, \qquad \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}, \qquad \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{pmatrix}.

Origin: Stated according to the original source

Problem 5.2.14

Let $A, B$ be hermitian matrices over $\mathbb {C}$ and eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$ and $\mu_{1}, \ldots , \mu_{n}$ , respectively. Assume that $\operatorname {tr}(AB)=0$ (scalar product). What can be said about the eigenvalues of $A+B$ ?

Origin: Stated according to the original source

Problem 5.2.15

Consider the reverse-diagonal $n \times n$ unitary matrix

A(\phi _{1},\ldots ,\phi _{n}) = \begin{pmatrix} 0 & 0 & \cdots & 0 & e^{i\phi _{1}} \\ 0 & 0 & \cdots & e^{i\phi _{2}} & 0 \\ \vdots & \vdots & & & \\ 0 & e^{i\phi _{n-1}} & & 0 & 0 \\ e^{i\phi _{n}} & 0 & & 0 & 0 \end{pmatrix}

where $\phi_{j} \in \mathbb {R}$ ( $j=1,\ldots ,n$ ). Find the eigenvalues and eigenvectors.

Origin: Stated according to the original source

Problem 5.2.16

(i)(ii)

Let $A, B$ be $n \times n$ matrices over $\mathbb {C}$ . The two $n \times n$ matrices $A$ and $B$ have a common eigenvector if and only if the $n \times n$ matrix

\sum _{j,k=1}^{n-1} \left[A^{j},B^{k}\right]^{*}\left[A^{j},B^{k}\right]

is singular, i.e. the determinant is equal to $0$ .

Origin: Stated according to the original source

(i)

Apply the theorem to the $2 \times 2$ matrices

A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad B = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}.

(ii)

Apply the theorem to the $3 \times 3$ matrices

A(\alpha ) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos (\alpha ) & -\sin (\alpha ) \\ 0 & \sin (\alpha ) & \cos (\alpha ) \end{pmatrix}, \qquad B(\alpha ) = \begin{pmatrix} \cos (\alpha ) & -\sin (\alpha ) & 0 \\ \sin (\alpha ) & \cos (\alpha ) & 0 \\ 0 & 0 & 1 \end{pmatrix}.

Problem 5.2.17

Let $0 \leq x < 1$ . Consider the $N \times N$ matrix $C$ (correlation matrix) with the entries $C_{jk} := x^{\left|j-k\right|}$ , $j,k=1,\ldots ,N$ . Find the eigenvalues of $C$ . Show that if $N \rightarrow \infty$ the distribution of its eigenvalues becomes a continuous function of $\phi \in \left[0,2\pi \right]$

\lambda (\phi ) = \frac{1-x^{2}}{1-2x\cos (\phi )+x^{2}}.

Origin: Stated according to the original source

Problem 5.2.18

(i)(ii)

Origin: Stated according to the original source

(i)

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . Show that $A$ is normal if and only if there exists an $n \times n$ unitary matrix $U$ and an $n \times n$ diagonal matrix $D$ such that $D=U^{-1}AU$ . Note that $U^{-1}=U^{*}$ .

(ii)

Let $A$ be a normal $n \times n$ matrix over $\mathbb {C}$ . Show that $A$ has a set of $n$ orthonormal eigenvectors. Show that if $A$ has a set of $n$ orthonormal eigenvectors, then $A$ is normal.

Problem 5.2.19

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

The $2n \times 2n$ symplectic matrix is defined by

S = \begin{pmatrix} 0_{n} & I_{n} \\ -I_{n} & 0_{n} \end{pmatrix}.

The matrix $S$ is unitary and skew-hermitian. Find the eigenvalues of $S$ from this information.

(ii)

Find the eigenvalues of the $2n \times 2n$ matrix

R = \frac{1}{\sqrt{2}}\begin{pmatrix} I_{n} & I_{n} \\ -I_{n} & I_{n} \end{pmatrix}.

(iii)

Let $A$ be an $n \times n$ matrix over $\mathbb {R}$ . Let $\lambda_{1}, \ldots , \lambda_{n}$ be the eigenvalues. What can be said about the eigenvalues of the $2n \times 2n$ matrix

\begin{pmatrix} 0_{n} & A \\ A^{T} & 0_{n} \end{pmatrix}?

Problem 5.2.20

Given the matrix

A = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{pmatrix}.

Prove or disprove that exactly two eigenvalues are $0$ .

Origin: Stated according to the original source

Problem 5.2.21

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . What is the condition on $A$ such that all eigenvalues are $0$ and $A$ admits only one eigenvector?

Origin: Stated according to the original source

Problem 5.2.22

Consider the $n \times n$ matrix

J_{n} = \begin{pmatrix} 0 & 1 & 0 & & \cdots & 0 \\ 0 & 0 & 1 & 0 & \cdots & 0 \\ & & \ddots & \ddots & & \\ & & & & & 1 \\ & & & & & 0 \end{pmatrix}.

Hence an arbitrary Jordan block is given by $zI_{n}+J_{n}$ , where $z \in \mathbb {C}$ . Find the eigenvalues of

\begin{pmatrix} a & b \\ -b & a \end{pmatrix} \otimes I_{n} + I_{2} \otimes J_{n}.

Origin: Stated according to the original source

Problem 5.2.23

(i)(ii)(iii)

Let $I_{2}$ be the $2 \times 2$ identity matrix, $0_{2}$ be the $2 \times 2$ zero matrix and $\sigma_{3}$ be the Pauli spin matrix.

Origin: Stated according to the original source

(i)

Find the eigenvalues and normalized eigenvectors of the $4 \times 4$ matrix

A(t) = \begin{pmatrix} \cosh (2t)I_{2} & \sinh (2t)\sigma _{3} \\ \sinh (2t)\sigma _{3} & \cosh (2t)I_{2} \end{pmatrix}.

(ii)

Find the eigenvalues and normalized eigenvectors of the $6 \times 6$ matrix

B(t) = \begin{pmatrix} \cosh (2t)I_{2} & 0_{2} & \sinh (2t)\sigma _{3} \\ 0_{2} & I_{2} & 0_{2} \\ \sinh (2t)\sigma _{3} & 0_{2} & \cosh (2t)I_{2} \end{pmatrix}.

Can the results from (i) be utilized here?

(iii)

Find the eigenvalues of the $6 \times 6$ matrices

\begin{pmatrix} \frac{1}{\sqrt{2}}I_{2} & 0_{2} & \frac{1}{\sqrt{2}}I_{2} \\ 0_{2} & I_{2} & 0_{2} \\ \frac{1}{\sqrt{2}}I_{2} & 0_{2} & -\frac{1}{\sqrt{2}}I_{2} \end{pmatrix}, \qquad \begin{pmatrix} 0_{3} & I_{3} \\ -I_{3} & 0_{3} \end{pmatrix}.

Problem 5.2.24

Let $n \geq 3$ . Consider the tridiagonal $n \times n$ matrix

A = \begin{pmatrix} a_{1} & b_{2} & 0 & & 0 & 0 \\ c_{2} & a_{2} & b_{3} & & 0 & 0 \\ 0 & c_{3} & a_{3} & & 0 & 0 \\ \vdots & \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & a_{n-1} & b_{n} \\ 0 & 0 & 0 & \cdots & c_{n} & a_{n} \end{pmatrix}

with $a_{1}, a_{j}, b_{j}, c_{j} \in \mathbb {C}$ ( $j=1,\ldots ,n$ ). It has in general $n$ complex eigenvalues the $n$ roots of the characteristic polynomial $p(\lambda )$ . Show that this polynomial can be evaluated by the recursive formula

\begin{align} p_{k}(\lambda ) & = (\lambda -a_{k})p_{k-1}-b_{k}c_{k}p_{k-2}(\lambda ), \qquad k=2,3,\ldots ,n \\ p_{1}(\lambda ) & = \lambda -a_{1} \\ p_{0}(\lambda ) & = 1. \end{align}

Origin: Stated according to the original source

Problem 5.2.25

Consider an $n \times n$ symmetric tridiagonal matrix over $\mathbb {R}$ . Let $f_{n}(\lambda ) := \operatorname {det}\left(A-\lambda I_{n}\right)$ and

f_{k}(\lambda ) = \det \begin{pmatrix} \alpha _{1}-\lambda & \beta _{1} & 0 & & & 0 \\ \beta _{1} & \alpha _{2}-\lambda & \beta _{2} & & & 0 \\ 0 & \beta _{2} & \ddots & & \cdots & 0 \\ \vdots & \vdots & \vdots & & & \vdots \\ 0 & \cdots & \cdots & & \alpha _{k-1}-\lambda & \beta _{k-1} \\ 0 & \cdots & 0 & & \beta _{k-1} & \alpha _{k}-\lambda \end{pmatrix}

for $k=1,2,\ldots ,n$ and $f_{0}(\lambda )=1$ , $f_{-1}(\lambda )=0$ . Then

f_{k}(\lambda ) = (\alpha -\lambda )f_{k-1}(\lambda )-\beta _{k-1}^{2}f_{k-2}(\lambda )

for $k=2,3,\ldots ,n$ . Find $f_{4}(\lambda )$ for the $4 \times 4$ matrix

\begin{pmatrix} 0 & \sqrt{1} & 0 & 0 \\ \sqrt{1} & 0 & \sqrt{2} & 0 \\ 0 & \sqrt{2} & 0 & \sqrt{3} \\ 0 & 0 & \sqrt{3} & 0 \end{pmatrix}.

Origin: Stated according to the original source

Problem 5.2.26

Let $\epsilon_{j} \in \mathbb {R}$ with $j=1,2,3,4$ . Consider the tridiagonal hermitian matrix

H(\epsilon _{j}) = \begin{pmatrix} \epsilon _{1} & 1 & 0 & 0 \\ 1 & \epsilon _{2} & 1 & 0 \\ 0 & 1 & \epsilon _{3} & 1 \\ 0 & 0 & 1 & \epsilon _{4} \end{pmatrix}.

Find the eigenvalues and eigenvectors. Let $\overrightarrow {v}=\begin{pmatrix} v_{1} & v_{2} & v_{3} & v_{4} \end{pmatrix}^{T}$ be an eigenvectors. Find $v_{1}/v_{2}$ , $v_{2}/v_{3}$ , $v_{3}/v_{4}$ .

Origin: Stated according to the original source

Problem 5.2.27

Let $M$ be an $n \times n$ invertible matrix with eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$ (counting degeneracy). Find the eigenvalues of

M_{+} = \frac{1}{2}(M+M^{-1}) \quad \text{and} \quad M_{-} = \frac{1}{2}(M-M^{-1}).

Origin: Stated according to the original source

Problem 5.2.28

Let $A=(a_{jk})$ be an $n \times n$ semi-positive definite matrix with $\sum_{k=1}^{n} a_{jk} < 1$ for $j=1,\ldots ,n$ . Show that $0 \leq \lambda < 1$ for all eigenvalues $\lambda$ of $A$ .

Origin: Stated according to the original source

Problem 5.2.29

(i)(ii)

Given an $n \times n$ permutation matrix $P$ .

Origin: Stated according to the original source

(i)

Assume that $n$ is even and $\operatorname {tr}(P)=0$ . Can we conclude that half of the eigenvalues of such a matrix are $+1$ and the other half are $-1$ ?

(ii)

Assume that $n$ is odd and $\operatorname {tr}(P)=0$ . Can we conclude that the eigenvalues are given by the $n$ solutions of $\lambda^{n}=1$ ?

Problem 5.2.30

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . The characteristic polynomial of $A$ is defined by

\Delta (\lambda ) := \operatorname {det}\left(\lambda I_{n}-A\right) = \lambda ^{n}+\sum _{k=1}^{n} (-1)^{k}\sigma (k)\lambda ^{n-k}.

The eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$ of $A$ are the solutions of the characteristic equation $\Delta (\lambda )=0$ . The coefficients $\sigma (j)$ ( $j=1,\ldots ,n$ ) are given by

\begin{align} \sigma (1) & = \sum _{j=1}^{n} \lambda _{j} = \operatorname {tr}(A) \\ \sigma (2) & = \sum _{j<k} \lambda _{j}\lambda _{k} \\ & \vdots \\ \sigma (n) & = \prod _{j=1}^{n} \lambda _{j} = \operatorname {det}\left(A\right). \end{align}

Another set of symmetric polynomials is given by the traces of powers of the matrix $A$ , namely

s(j) = \sum _{k=1}^{n} (\lambda _{k})^{j} = \operatorname {tr}(A^{j}), \qquad j=1,\ldots ,n.

One has (so-called Newton relation)

j\sigma (j)-s(1)\sigma (j-1)+\cdots +(-1)^{j-1}s(j-1)\sigma (1)+(-1)^{j}s(j) = 0

where $j=1,\ldots ,n$ . Consider the $3 \times 3$ matrix

A = \begin{pmatrix} 1/\sqrt{2} & 0 & 1/\sqrt{2} \\ 0 & 1 & 0 \\ 1/\sqrt{2} & 0 & -1/\sqrt{2} \end{pmatrix}.

Calculate $s(1), s(2), s(3)$ from the traces of the powers of $A$ . Then apply the Newton relation to find $\sigma (1), \sigma (2), \sigma (3)$ .

Origin: Stated according to the original source

Problem 5.2.31

(i)(ii)(iii)(iv)

Origin: Stated according to the original source

(i)

Is the $3 \times 3$ stochastic matrix

\begin{pmatrix} 1/3 & 1/3 & 1/3 \\ 2/3 & 0 & 1/3 \\ 1/2 & 1/2 & 0 \end{pmatrix}

diagonalizable? First find the eigenvalues and normalized eigenvectors.

(ii)

Find the eigenvalues and eigenvectors of the stochastic matrix

\begin{pmatrix} 1/2 & 1/2 & 0 \\ 0 & 1/2 & 1/2 \\ 1 & 0 & 0 \end{pmatrix}.

(iii)

Find the eigenvalues and eigenvectors of the stochastic matrix

\begin{pmatrix} p_{AA} & p_{AB} & p_{AC} & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ p_{DA} & p_{DB} & p_{DC} & 0 \end{pmatrix}

where $p_{AA}=p_{DA}=2-\sqrt{3}$ , $p_{AB}=p_{DB}=\sqrt{3}-\sqrt{2}$ , $p_{AC}=p_{DC}=\sqrt{2}-1$ .

(iv)

Find the eigenvalues of the double stochastic matrices

\begin{pmatrix} \sin ^{2}(\theta ) & \cos ^{2}(\theta ) \\ \cos ^{2}(\theta ) & \sin ^{2}(\theta ) \end{pmatrix}, \qquad \begin{pmatrix} \sin ^{2}(\theta ) & 0 & \cos ^{2}(\theta ) \\ \cos ^{2}(\theta ) & \sin ^{2}(\theta ) & 0 \\ 0 & \cos ^{2}(\theta ) & \sin ^{2}(\theta ) \end{pmatrix},

\begin{pmatrix} \sin ^{2}(\theta ) & 0 & 0 & \cos ^{2}(\theta ) \\ \cos ^{2}(\theta ) & \sin ^{2}(\theta ) & 0 & 0 \\ 0 & \cos ^{2}(\theta ) & \sin ^{2}(\theta ) & 0 \\ 0 & 0 & \cos ^{2}(\theta ) & \sin ^{2}(\theta ) \end{pmatrix}.

Problem 5.2.32

(i)(ii)

Origin: Stated according to the original source

(i)

Let $c_{j} \in \mathbb {R}$ . Find the eigenvalues of the matrices

\begin{pmatrix} 0 & 1 \\ c_{1} & c_{2} \end{pmatrix}, \qquad \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ c_{1} & c_{2} & c_{3} \end{pmatrix}, \qquad \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ c_{1} & c_{2} & c_{3} & c_{4} \end{pmatrix}.

Generalize to the $n \times n$ case. Note that for the $2 \times 2$ matrix we find

\lambda _{1,2} = \frac{c_{2}}{2} \pm \frac{1}{2}\sqrt{4c_{1}+c_{2}^{2}}.

(ii)

Show that the eigenvalues the $4 \times 4$ matrix

A = \begin{pmatrix} 0 & a_{12} & a_{13} & a_{14} \\ a_{12} & 0 & 0 & 0 \\ a_{13} & 0 & 0 & 0 \\ a_{14} & 0 & 0 & 0 \end{pmatrix}

are given by $0$ (2 times) and $\sqrt{a_{12}^{2}+a_{13}^{2}+a_{14}^{2}}, -\sqrt{a_{12}^{2}+a_{13}^{2}+a_{14}^{2}}$ . Find the eigenvectors.

Problem 5.2.33

Consider the $3 \times 3$ matrices

A = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix}.

Find the eigenvalues of $A$ and $B$ and the eigenvalues of the commutator $\left[A,B\right]$ . Are the matrices $A$ and $B$ similar?

Origin: Stated according to the original source

Problem 5.2.34

What can be said about the eigenvalues and eigenvectors of a nonzero hermitian $n \times n$ matrix $A$ with $\operatorname {det}\left(A\right)=0$ and $\operatorname {tr}(A)=0$ . Give one eigenvalue of $A$ . Give one eigenvalue of $A \otimes A$ . Consider the case $n=3$ for the matrix $A$ . Find all the eigenvalues and eigenvectors.

Origin: Stated according to the original source

Problem 5.2.35

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . Show that the matrix $A$ admits the inverse matrix $A^{-1}$ iff $A\overrightarrow {v}=\overrightarrow {0}$ implies $\overrightarrow {v}=\overrightarrow {0}$ . Note that from $A\overrightarrow {v}=\overrightarrow {0}$ it follows that $A^{-1}(A\overrightarrow {v})=\overrightarrow {0}$ and then $\overrightarrow {v}=\overrightarrow {0}$ .

Origin: Stated according to the original source

Problem 5.2.36

Let $S$ and $Q$ be $n \times n$ matrices and

(S-\lambda I_{n})\overrightarrow {u} = \overrightarrow {0}, \qquad (Q-\mu I_{n})\overrightarrow {v} = \overrightarrow {0}.

These are eigenvalue equations. Show that

(S \otimes Q - \lambda \mu (I_{n} \otimes I_{n}))(\overrightarrow {u} \otimes \overrightarrow {v}) = \overrightarrow {0}_{n^{2}}

and

(S \oplus Q - \lambda \mu (I_{n} \oplus I_{n}))\begin{pmatrix} \overrightarrow {u} \\ \overrightarrow {v} \end{pmatrix} = \overrightarrow {0}_{2n}.

Origin: Stated according to the original source

Problem 5.2.37

Let $\gamma \in \mathbb {R}$ and $\gamma^{2}<1$ . Consider the $2 \times 2$ matrix

K = \sigma _{1}-i\gamma \sigma _{3}.

Show that the eigenvalues of $K$ and $K^{*}$ are given by $\pm \sqrt{1-\gamma^{2}}$ . Find the normalized eigenvectors.

Origin: Stated according to the original source

Problem 5.2.38

Let $n \geq 2$ . Consider the $(n-1) \times n$ matrix $A$ over $\mathbb {R}$ . Then $AA^{T}$ is an $(n-1) \times (n-1)$ matrix over $\mathbb {R}$ and $A^{T}A$ is an $n \times n$ matrix over $\mathbb {R}$ . Show that the $(n-1)$ eigenvalues of $AA^{T}$ are also eigenvalues of $A^{T}A$ and $A^{T}A$ additionally admits the eigenvalue $0$ .

Origin: Stated according to the original source

Problem 5.2.39

Let $A, B$ be $3 \times 3$ matrices. We define the composition

A \diamond B := \begin{pmatrix} a_{11} & 0 & a_{12} & 0 & a_{13} \\ 0 & b_{11} & b_{12} & b_{13} & 0 \\ a_{21} & b_{21} & a_{22}b_{22} & b_{23} & a_{23} \\ 0 & b_{31} & b_{32} & b_{33} & 0 \\ a_{31} & 0 & a_{32} & 0 & a_{33} \end{pmatrix}.

Let

M = \begin{pmatrix} 1/\sqrt{2} & 0 & 1/\sqrt{2} \\ 0 & 1 & 0 \\ 1/\sqrt{2} & 0 & -1/\sqrt{2} \end{pmatrix}.

Find the eigenvalues of $M$ and $M \diamond M$ .

Origin: Stated according to the original source

Problem 5.2.40

Let $j,k \in \left\{ 0,1\right\}$ . Find the eigenvalues of the $2 \times 2$ matrix $M=(M_{jk})$ with $M_{jk}=e^{i\pi (j \cdot k)}$ .

Origin: Stated according to the original source

Problem 5.2.41

Compare the lowest eigenvalues of the two $6 \times 6$ matrices

\begin{pmatrix} 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{pmatrix}, \qquad \begin{pmatrix} 0 & 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 0 & 1 & 0 \end{pmatrix}.

Origin: Stated according to the original source

Problem 5.2.42

(i)(ii)(iii)(iv)

Let $A_{n}$ be the $n \times n$ matrices of the form

A_{2} = \begin{pmatrix} 0 & t \\ t & r \end{pmatrix}, \qquad A_{3} = \begin{pmatrix} 0 & 0 & t \\ 0 & 1 & 0 \\ t & 0 & r \end{pmatrix}, \qquad A_{4} = \begin{pmatrix} 0 & 0 & 0 & t \\ 0 & 0 & t & 0 \\ 0 & t & r & 0 \\ t & 0 & 0 & r \end{pmatrix},

A_{5} = \begin{pmatrix} 0 & 0 & 0 & 0 & t \\ 0 & 0 & 0 & t & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & t & 0 & r & 0 \\ t & 0 & 0 & 0 & r \end{pmatrix}.

Thus the even dimensional matrix $A_{2n}$ has $t$ along the skew-diagonal and $r$ along the lower main diagonal. Otherwise the entries are $0$ . The odd dimensional matrix $A_{2n+1}$ has $t$ along the skew-diagonal except $1$ at the centre and $r$ along the lower main diagonal. Otherwise the entries are $0$ . Find the eigenvalues of these matrices.

Origin: Stated according to the original source

(i)

Show that $n=2$ the eigenvalues are

-\frac{1}{2}\left[\sqrt{4t^{2}+r^{2}}-r\right], \qquad \frac{1}{2}\left[\sqrt{4t^{2}+r^{2}}+r\right].

(ii)

Show that $n=3$ the eigenvalues are

-\frac{1}{2}\left[\sqrt{4t^{2}+r^{2}}-r\right], \qquad \frac{1}{2}\left[\sqrt{4t^{2}+r^{2}}+r\right], \qquad 1.

(iii)

Show that for $n=4$ the eigenvalues are

-\frac{1}{2}\left[\sqrt{4t^{2}+r^{2}}-r\right], \qquad \frac{1}{2}\left[\sqrt{4t^{2}+r^{2}}+r\right]

twice each.

(iv)

Show that for $n=5$ we have the same eigenvalues as for $n=4$ and additional the eigenvalue $1$ . Extend to general $n$ .

Problem 5.2.43

Let $n \geq 2$ and $j,k=0,1,\ldots ,n-1$ . Consider

M(j,k) = \begin{cases} 1, & \text{for } j<k, \\ 0, & \text{for } j=k, \\ -1, & \text{for } j>k. \end{cases}

Write down the matrix $M$ for $n=2$ and find the eigenvalues and normalized eigenvectors. Write down the matrix $M$ for $n=3$ and find the eigenvalues and normalized eigenvectors.

Origin: Stated according to the original source

Problem 5.2.44

Find the spectrum of the $3 \times 3$ matrix $A=(a_{jk})$ ( $j,k=1,2,3$ )

a_{jk} = \sqrt{j(k-1)}.

Origin: Stated according to the original source

Problem 5.2.45

(i)(ii)

Let $U$ be a unitary $4 \times 4$ matrix. Assume that each column vector (which obviously is normalized) of $U$ can be written as the Kronecker product of two normalized vectors in $\mathbb {C}^{2}$ .

Origin: Stated according to the original source

(i)

Can we conclude that the eigenvectors of such a unitary matrix can also be written a Kronecker product of two vectors in $\mathbb {C}^{2}$ ?

(ii)

Can the matrix $U$ be written as a Kronecker product of two $2 \times 2$ matrices?

Problem 5.2.46

Let $A_{1}$ be an $n_{1} \times n_{1}$ matrix, $A_{2}$ be an $n_{2} \times n_{2}$ matrix, $A_{3}$ be an $n_{3} \times n_{3}$ matrix and $I_{n_{j}}$ be the $n_{j} \times n_{j}$ matrices ( $j=1,2,3$ ). Let $\overrightarrow {u}$ be a normalized eigenvector of $A_{1}$ with eigenvalue $\lambda$ , $\overrightarrow {v}$ be a normalized eigenvector of $A_{2}$ with eigenvalue $\mu$ and $\overrightarrow {w}$ be a normalized eigenvector with eigenvalue $\nu$ . Consider

\begin{align} \hat{H} & = A_{1} \otimes I_{n_{2}} \otimes I_{n_{3}} + I_{n_{1}} \otimes A_{2} \otimes I_{n_{3}} + I_{n_{1}} \otimes I_{n_{2}} \otimes A_{3} \\ & \quad + A_{1} \otimes A_{2} \otimes I + A_{1} \otimes I \otimes A_{3} + I \otimes A_{2} \otimes A_{3} + A_{1} \otimes A_{2} \otimes A_{3}. \end{align}

Show that $\overrightarrow {u} \otimes \overrightarrow {v} \otimes \overrightarrow {w}$ is a normalized eigenvector of $\hat{H}$ . Find the corresponding eigenvalue.

Origin: Stated according to the original source

Problem 5.2.47

Let $\lambda$ be an eigenvalue of the $n \times n$ matrix $A$ with normalized eigenvector $\overrightarrow {v}$ and $\mu$ be an eigenvalue of the $n \times n$ matrix $B$ with normalized eigenvector $\overrightarrow {w}$ . Show that $\lambda \mu -\lambda -\mu$ is an eigenvalue of $A \otimes B - A \otimes I_{n} - I_{n} \otimes B$ with normalized eigenvector $\overrightarrow {v} \otimes \overrightarrow {w}$ .

Origin: Stated according to the original source

Problem 5.2.48

(i)(ii)

Let $A, B$ be $2 \times 2$ matrices over $\mathbb {C}$ . Let $\lambda$ be an eigenvalue of $A$ with corresponding normalized eigenvector $\overrightarrow {u}$ and $\mu$ be an eigenvalue of $B$ with corresponding normalized eigenvector $\overrightarrow {v}$ . Let $A \otimes B$ be the Kronecker product of $A$ and $B$ . Then $\lambda \mu$ is an eigenvalue of $A \otimes B$ with corresponding normalized eigenvector $\overrightarrow {u} \otimes \overrightarrow {v}$ . Let $A \oplus B$ be the direct sum of $A$ and $B$ . Then we have

\begin{pmatrix} A & 0_{2} \\ 0_{2} & B \end{pmatrix}\begin{pmatrix} u_{1} \\ u_{2} \\ 0 \\ 0 \end{pmatrix} = \lambda \begin{pmatrix} u_{1} \\ u_{2} \\ 0 \\ 0 \end{pmatrix}, \qquad \begin{pmatrix} A & 0_{2} \\ 0_{2} & B \end{pmatrix}\begin{pmatrix} 0 \\ 0 \\ v_{1} \\ v_{2} \end{pmatrix} = \mu \begin{pmatrix} 0 \\ 0 \\ v_{1} \\ v_{2} \end{pmatrix}.

Consequently

\begin{pmatrix} A & 0_{2} \\ 0_{2} & B \end{pmatrix}\begin{pmatrix} u_{1} \\ u_{2} \\ v_{1} \\ v_{2} \end{pmatrix} = \begin{pmatrix} \lambda u_{1} \\ \lambda u_{2} \\ \mu v_{1} \\ \mu v_{2} \end{pmatrix}.

Thus the vector $\begin{pmatrix} u_{1} & u_{2} & v_{1} & v_{2} \end{pmatrix}^{T}$ would be an eigenvector of $A \oplus B$ if $\lambda =\mu$ . Let $A \star B$ be the star product

A \star B = \begin{pmatrix} b_{11} & 0 & 0 & b_{12} \\ 0 & a_{11} & a_{12} & 0 \\ 0 & a_{21} & a_{22} & 0 \\ b_{21} & 0 & 0 & b_{22} \end{pmatrix} \quad \Rightarrow \quad (A \star B)\begin{pmatrix} v_{1} \\ u_{1} \\ u_{2} \\ v_{2} \end{pmatrix} = \begin{pmatrix} \mu v_{1} \\ \lambda u_{1} \\ \lambda u_{2} \\ \mu v_{2} \end{pmatrix}.

Thus the vector $\begin{pmatrix} v_{1} & u_{1} & u_{2} & v_{2} \end{pmatrix}^{T}$ would be an eigenvector of $A \star B$ if $\mu =\lambda$ .

Origin: Stated according to the original source

(i)

Study the eigenvalue problem for $A \otimes B + A \oplus B + A \star B$ .

(ii)

Apply the result to $A=\sigma_{1}$ and $B=\sigma_{2}$ , where $\sigma_{1}, \sigma_{2}$ denote the Pauli spin matrices. Both admit the eigenvalues $+1, -1$ .

Problem 5.2.49

(i)(ii)

Origin: Stated according to the original source

(i)

Given the eigenvalues of the $n \times n$ matrix $M$ . Find the eigenvalues of

e^{-M} \otimes I_{n} + I_{n} \otimes e^{M}.

(ii)

Let $A$ and $B$ be normal $n \times n$ matrices with eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$ and $\mu_{1}, \ldots , \mu_{n}$ , respectively. Find the eigenvalues of $e^{i\phi }A \otimes I_{n}+e^{-i\phi }I_{n} \otimes B$ .

Problem 5.2.50

Let $\alpha \in \mathbb {R}$ . Consider the symmetric $4 \times 4$ matrix

A(\alpha ) = \frac{1}{\sqrt{2}}\begin{pmatrix} 1+\alpha & 0 & 1-\alpha & 0 \\ 0 & 1+\alpha & 0 & 1-\alpha \\ 1-\alpha & 0 & 1+\alpha & 0 \\ 0 & 1-\alpha & 0 & 1+\alpha \end{pmatrix}.

Show that an invertible matrix $B$ and diagonal matrix $D$ such that $A(\alpha )=B^{-1}DB$ are given by

B = B^{-1} = \frac{1}{2}\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 \end{pmatrix}, \qquad D = \sqrt{2}\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \alpha & 0 \\ 0 & 0 & 0 & \alpha \end{pmatrix}.

Origin: Stated according to the original source

Problem 5.2.51

Let $A$ be an $n \times n$ matrix with eigenvalue $\lambda$ and eigenvector $\overrightarrow {u}$ , i.e. $A\overrightarrow {u}=\lambda \overrightarrow {u}$ , $\overrightarrow {u} \neq \overrightarrow {0}$ . Consider the $n^{n} \times n^{n}$ matrix

K = A \otimes I_{n} \otimes \cdots \otimes I_{n} + I_{n} \otimes A \otimes I_{n} \otimes \cdots \otimes I_{n} + \cdots + I_{n} \otimes \cdots \otimes I_{n} \otimes A

where each term has $n$ Kronecker products. Show that

K(\overrightarrow {u} \otimes \cdots \otimes \overrightarrow {u}) = \lambda (\overrightarrow {u} \otimes \cdots \otimes \overrightarrow {u}) + \cdots + \lambda (\overrightarrow {u} \otimes \cdots \otimes \overrightarrow {u}) = n\lambda (\overrightarrow {u} \otimes \cdots \otimes \overrightarrow {u})

i.e. $n\lambda$ is an eigenvalue.

Origin: Stated according to the original source

Problem 5.2.52

Consider the $(n+1) \times (n+1)$ matrix

A = \begin{pmatrix} 0 & \overrightarrow {s}^{*} \\ \overrightarrow {r} & 0_{n \times n} \end{pmatrix}

where $\overrightarrow {r}$ and $\overrightarrow {s}$ are $n \times 1$ vectors with complex entries, $\overrightarrow {s}^{*}$ denoting the conjugate transpose of $\overrightarrow {s}$ . Show that the characteristic polynomial is given by

\operatorname {det}\left(B-\lambda I_{n+1}\right) = (-\lambda )^{n-1}(\lambda ^{2}-\overrightarrow {s}^{*}\overrightarrow {r}).

Origin: Stated according to the original source

Problem 5.2.53

Let $n \geq 2$ and $n$ even. Find the eigenvalues of the $n \times n$ matrices

A = \begin{pmatrix} 2 & -1 & 0 & & 0 & -1 \\ -1 & 2 & -1 & & & 0 \\ 0 & \ddots & \ddots & & & \vdots \\ \vdots & & & & & 0 \\ 0 & & \ddots & \ddots & & -1 \\ -1 & 0 & & 0 & -1 & 2 \end{pmatrix}.

Origin: Stated according to the original source

Problem 5.2.54

Let $j,k=0,1,2,3$ . Write down the $4 \times 4$ matrix $A$ with the entries

a_{jk} = \cos \left[\frac{\pi j-\pi k}{2}\right]

and find the eigenvalues.

Origin: Stated according to the original source

Problem 5.2.55

(i)(ii)(iii)(iv)

Let $H$ be a hermitian $n \times n$ matrix and $U$ an $n \times n$ unitary matrix such that $UHU^{*}=H$ . We call $H$ invariant under $U$ . From $UHU^{*}=H$ it follows that $\left[H,U\right]=0_{n}$ . If $\overrightarrow {v}$ is an eigenvector of $H$ , i.e. $H\overrightarrow {v}=\lambda \overrightarrow {v}$ , then $U\overrightarrow {v}$ is also an eigenvector of $H$ since

H(U\overrightarrow {v}) = (HU)\overrightarrow {v} = U(H\overrightarrow {v}) = \lambda (U\overrightarrow {v}).

The set of all unitary matrices $U_{j}$ ( $j=1,\ldots ,m$ ) that leave a given hermitian matrix invariant, i.e. $U_{j}HU_{j}^{*}=H$ ( $j=1,\ldots ,m$ ) form a group under matrix multiplication.

Origin: Stated according to the original source

(i)

Find all $2 \times 2$ hermitian matrices $H$ such that $\left[H,U\right]=0_{2}$ , where $U=\sigma_{1}$ .

(ii)

Find all $2 \times 2$ hermitian matrices $H$ such that $\left[H,U\right]=0_{2}$ , where $U=\sigma_{2}$ .

(iii)

Find all $4 \times 4$ hermitian matrices $H$ such that $\left[H,U\right]=0_{4}$ , where $U=\sigma_{1} \otimes \sigma_{1}$ .

(iv)

Find all $4 \times 4$ hermitian matrices $H$ such that $\left[H,U\right]=0_{4}$ , where $U=\sigma_{2} \otimes \sigma_{2}$ .

Problem 5.2.56

(i)(ii)(iii)

The Cartan matrix $A$ of a rank $r$ root system of a semi-simple Lie algebra is an $r \times r$ matrix whose entries are derived from the simple roots. The entries of the Cartan matrix are given by

a_{jk} = 2\frac{(\alpha _{j},\alpha _{k})}{\alpha _{j},\alpha _{j}}

where $(\, , \, )$ is the Euclidean inner product and $\alpha_{j}$ are the simple roots. The entries are independent of the choice of simple roots (up to ordering).

Origin: Stated according to the original source

(i)

The Cartan matrix of the Lie algebra $e_{8}$ is given by the $8 \times 8$ matrix

\begin{pmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 2 \end{pmatrix}.

Show that the trace is equal to $16$ . Show that the determinant is equal to $1$ . Find the eigenvalues and eigenvectors of the matrix.

(ii)

The Lie algebra $\mathfrak {sl}(3,\mathbb {R})$ is the rank $2$ Lie algebra with Cartan matrix

C = \begin{pmatrix} 2 & -1 \\ 1 & 2 \end{pmatrix}.

Find the eigenvalues and normalized eigenvectors of $C$ .

(iii)

Let $\tau =2\cos (\pi /5)$ . Find the eigenvalues and eigenvectors of the three Cartan matrices

A_{2} = \begin{pmatrix} 2 & -\tau \\ -\tau & 2 \end{pmatrix}, \qquad A_{3} = \begin{pmatrix} 2 & -1 & 0 \\ -1 & 2 & -\tau \\ 0 & -\tau & 2 \end{pmatrix}, \qquad A_{4} = \begin{pmatrix} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -\tau \\ 0 & 0 & -\tau & 2 \end{pmatrix}.

Problem 5.2.57

Let $A$ be an $n \times n$ matrix. Let $J$ be the $n \times n$ matrix with $1$ 's in the counter diagonal and $0$ 's otherwise. Let $\operatorname {tr}(A)=0$ , $\operatorname {tr}(JA)=0$ . What can be said about the eigenvalues of $A$ ? Consider first the cases $n=2$ and $n=3$ .

Origin: Stated according to the original source

Problem 5.2.58

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . Let $n \geq 1$ . We define

\Delta (A^{n}) := \sum _{j=0}^{n} \binom {n}{j} A^{k} \otimes A^{n-k}.

Given the eigenvalues of $A$ . What can be said about the eigenvalues of $\Delta (A^{n})$ ?

Origin: Stated according to the original source

Problem 5.2.59

Let $A, B$ be $n \times n$ matrices over $\mathbb {C}$ . Assume we know the eigenvalues and eigenvectors of $A$ and $B$ . What can be said about the eigenvalues and eigenvectors of $A \otimes B - B \otimes A$ and $A \otimes B + B \otimes A$ ?

Origin: Stated according to the original source

Problem 5.2.60

Let $U$ be a unitary $4 \times 4$ matrix. Assume that each column (vector) can be written as the Kronecker product of two vectors in $\mathbb {C}^{2}$ . Can we conclude that the eigenvectors of such a unitary matrix can also be written a Kronecker product of two vectors in $\mathbb {C}^{2}$ ?

Origin: Stated according to the original source

Problem 5.2.61

Let $A, B$ be $n \times n$ normal matrices. Consider the normal matrices

H = A \otimes I_{n} + I_{n} \otimes B + A \otimes B, \qquad K = A \otimes I_{n} + I_{n} \otimes B + B \otimes A.

Given the eigenvalues and eigenvectors of $A$ , i.e. $\lambda_{j}, \overrightarrow {u}_{j}$ ( $j=1,\ldots ,n$ ) and the eigenvalues and eigenvectors of $B$ , i.e. $\mu_{j}, \overrightarrow {v}_{j}$ ( $j=1,\ldots ,n$ ). Then the eigenvalues and eigenvectors of $H$ are given by $\lambda_{j}+\mu_{k}+\lambda_{j}\mu_{k}$ , $\overrightarrow {u}_{j} \otimes \overrightarrow {v}_{k}$ . Assume that

\operatorname {tr}(AB^{*}) = 0

i.e. considering the Hilbert space of the $n \times n$ matrices with the scalar product $\left\langle X, Y \right\rangle := \operatorname {tr}(XY^{*})$ the matrices $A$ and $B$ are orthogonal. What can be said about the eigenvalues and eigenvectors of $K$ ?

Origin: Stated according to the original source

Problem 5.2.62

(i)(ii)

Origin: Stated according to the original source

(i)

Let $a,b,c \in \mathbb {R}$ . Show that the four eigenvalues of the $4 \times 4$ matrix

A = \begin{pmatrix} 0 & -a & -b & -c \\ a & 0 & -ic & ib \\ b & ic & 0 & -ia \\ c & -ib & ia & 0 \end{pmatrix}

are given by $\pm \sqrt{a^{2}+b^{2}+c^{2}}, \pm i\sqrt{a^{2}+b^{2}+c^{2}}$ .

(ii)

Let $x \in \mathbb {R}$ . Show that the lowest eigenvalue of the symmetric $4 \times 4$ matrix ( $x \in \mathbb {R}$ )

\begin{pmatrix} 0 & -x\sqrt{5} & 0 & 0 \\ -x\sqrt{5} & 4 & -2x & -2x \\ 0 & -2x & 4-2x & -x \\ 0 & -2x & -x & 8-2x \end{pmatrix}

is given by

\lambda _{0} = 4-x-2\sqrt{1+x^{2}+2x\cos (2\pi /5)}-2\sqrt{1+x^{2}+2x\cos (4\pi /5)}.

Problem 5.2.63

Let $A$ be an $n \times n$ normal matrix, i.e. $AA^{*}=A^{*}A$ . Let $\overrightarrow {u}$ be an eigenvector of $A$ , i.e. $A\overrightarrow {u}=\lambda \overrightarrow {u}$ . Show that $\overrightarrow {u}$ is also an eigenvector of $A^{*}$ with eigenvalue $\overline{\lambda }$ , i.e.

A^{*}\overrightarrow {u} = \overline{\lambda }\overrightarrow {u}.

Apply $(A\overrightarrow {u})^{*}A\overrightarrow {u} = (A^{*}\overrightarrow {u})^{*}A^{*}\overrightarrow {u}$ .

Origin: Stated according to the original source

Problem 5.2.64

Let $A$ be an $n \times n$ normal matrix. Assume that $\lambda_{j}$ ( $j=1,\ldots ,n$ ) are the eigenvalues of $A$ . Calculate

\prod _{k=1}^{n} (1+\lambda _{k})

without using the eigenvalues. Apply the identity

\prod _{k=1}^{n} (1+\lambda _{k}) \equiv \operatorname {det}\left(I_{n}+A\right).

Origin: Stated according to the original source

Problem 5.2.65

(i)(ii)(iii)(iv)(v)(vi)

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ , which satisfies $A^{2} \equiv AA = cA$ , where $c \in \mathbb {C}$ is a constant. Obviously the equation is satisfied by the zero matrix. Assume that $A \neq 0_{n}$ . Then we have a "type of eigenvalue equation".

Origin: Stated according to the original source

(i)

Is $c$ an eigenvalue of $A$ ?

(ii)

Take the determinant of both sides of the equation. Discuss. Study the cases that $A$ is invertible and non-invertible. If the matrix $A$ is invertible it follows that $A=cI_{n}$ and $c \neq 0$ . Taking the determinant yields

(\operatorname {det}\left(A\right))^{2} = c^{n}\operatorname {det}\left(A\right).

(iii)

Study the case

A(z) = \begin{pmatrix} e^{-z} & 1 \\ 1 & e^{z} \end{pmatrix}, \qquad z \in \mathbb {C}

and show that

A^{2}(z) = (e^{z}+e^{-z})A(z)

where $e^{z}+e^{-z}$ together with $0$ are the eigenvalues of $A(z)$ .

(iv)

Study $(A \otimes A)^{2}=c(A \otimes A)$ .

(v)

Let $A$ be a $2 \times 2$ matrix and $A \star A$ be the star product. Study the case $(A \star A)^{2}=c(A \star A)$ .

(vi)

Study the case that $A^{3}=cA$ .

Problem 5.2.66

The additive inverse eigenvalue problem is as follows: Let $A$ be an $n \times n$ symmetric matrix over $\mathbb {R}$ with $a_{jj}=0$ for $j=1,\ldots ,n$ . Find a real diagonal $n \times n$ matrix $D$ such that the matrix $A+D$ has the prescribed eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$ . The number of solutions for the real matrix $D$ varies from $0$ to $n!$ . Consider the $2 \times 2$ matrix $A=\sigma_{1}$ and the prescribed eigenvalues $\lambda_{1}=2$ , $\lambda_{2}=3$ . Can one find a $D$ ? We have

A+D = \begin{pmatrix} d_{11} & 1 \\ 1 & d_{22} \end{pmatrix}.

Origin: Stated according to the original source

Problem 5.2.67

Let $A$ be an $n \times n$ normal matrix with pairwise different eigenvalues. Are the matrices

\Pi _{j} = \prod _{k=1,j \neq k}^{n} \frac{A-\lambda _{k}I_{n}}{\lambda _{j}-\lambda _{k}}

projection matrices?

Origin: Stated according to the original source

Problem 5.2.68

Consider the hermitian $3 \times 3$ matrix

A = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix}.

Find $A^{2}$ and $A^{3}$ . We know that

\operatorname {tr}(A) = \lambda _{1}+\lambda _{2}+\lambda _{3}, \qquad \operatorname {tr}(A^{2}) = \lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}, \qquad \operatorname {tr}(A^{3}) = \lambda _{1}^{3}+\lambda _{2}^{3}+\lambda _{3}^{3}.

Use Newton's method to solve this system of three equations to find the eigenvalues of $A$ .

Origin: Stated according to the original source

Problem 5.2.69

Consider an $n \times n$ permutation matrix $P$ . Obviously $+1$ is always an eigenvalue since the column vector with all $n$ entries equal to $+1$ is an eigenvector. Apply a brute force method and give a C++ implementation to figure out whether $-1$ is an eigenvalue. We run over all column vectors $\overrightarrow {v}$ of length $n$ , where the entries can only be $+1$ or $-1$ , where of course the cases with all entries $+1$ or all entries $-1$ can be omitted. Thus the number of column vectors we have to run through are $2^{n}-2$ . The condition then to be checked is $P\overrightarrow {v}=-\overrightarrow {v}$ . If true we have an eigenvalues $-1$ with the corresponding eigenvector $\overrightarrow {v}$ .

Origin: Stated according to the original source

Problem 5.2.70

(i)(ii)(iii)

The power method is the simplest algorithm for computing eigenvectors and eigenvalues. Consider the vector space $\mathbb {R}^{n}$ with the Euclidean norm $\left\| \overrightarrow {x}\right\|$ of a vector $\overrightarrow {x} \in \mathbb {R}^{n}$ . The iteration is as follows: Given a nonsingular $n \times n$ matrix $M$ and a vector $\overrightarrow {x}_{0}$ with $\left\| \overrightarrow {x}_{0}\right\| =1$ . One defines

\overrightarrow {x}_{t+1} = \frac{M\overrightarrow {x}_{t}}{\left\| M\overrightarrow {x}_{t}\right\| }, \qquad t=0,1,\ldots

This defines a dynamical system on the sphere $S^{n-1}$ . Since $M$ is invertible we have

\overrightarrow {x}_{t} = \frac{M^{-1}\overrightarrow {x}_{t+1}}{\left\| M^{-1}\overrightarrow {x}_{t+1}\right\| }, \qquad t=0,1,\ldots

Origin: Stated according to the original source

(i)

Apply the power method to the nonnormal matrix

A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \quad \text{and} \quad \overrightarrow {x}_{0} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}.

(ii)

Apply the power method to the Bell matrix

B = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & -1 & 0 \\ 1 & 0 & 0 & -1 \end{pmatrix} \quad \text{and} \quad \overrightarrow {x}_{0} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}.

(iii)

Consider the $3 \times 3$ symmetric matrix over $\mathbb {R}$

A = \begin{pmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{pmatrix}.

Find the largest eigenvalue and the corresponding eigenvector using the power method. Start from the vector

\overrightarrow {v} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \quad \Rightarrow \quad A\overrightarrow {v} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \qquad A^{2}\overrightarrow {v} = \begin{pmatrix} 2 \\ -2 \\ 2 \end{pmatrix}.

The largest eigenvalue is $\lambda =2+\sqrt{2}$ with the eigenvector $\begin{pmatrix} 1 & -\sqrt{2} & 1 \end{pmatrix}^{T}$ .

Problem 5.2.71

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . Then any eigenvalue of $A$ satisfies the inequality

\left|\lambda \right| \leq \max _{1 \leq j \leq n} \sum _{k=1}^{n} \left|a_{jk}\right|.

Write a C++ program that calculates the right-hand side of the inequality for a given matrix. Apply the complex class of STL. Apply it to the matrix

A = \begin{pmatrix} i & 0 & 0 & i \\ 0 & 2i & 2i & 0 \\ 0 & 3i & 3i & 0 \\ 4i & 0 & 0 & 4i \end{pmatrix}.

Origin: Stated according to the original source