Math Problems Vault

Problem 5.2.1

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(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)

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(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)

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(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

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

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(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)

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(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.

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Problem 5.2.8

(i)(ii)

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(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?

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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?

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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}?

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Problem 5.2.12

(i)(ii)

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(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}.

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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 $A+B$ ?

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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.

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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 $0$ .

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(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}}.

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Problem 5.2.18

(i)(ii)

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(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)

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(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 $0$ .

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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?

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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}.

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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.

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(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}

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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}.

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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 $v_{1}/v_{2}$ , $v_{2}/v_{3}$ , $v_{3}/v_{4}$ .

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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}).

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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 $A$ .

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Problem 5.2.29

(i)(ii)

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

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(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 $\sigma (1), \sigma (2), \sigma (3)$ .

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Problem 5.2.31

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

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(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)

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(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?

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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.

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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 $\overrightarrow {v}=\overrightarrow {0}$ .

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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}.

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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.

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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 $0$ .

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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 $M \diamond M$ .

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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 $M_{jk}=e^{i\pi (j \cdot k)}$ .

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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}.

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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.

?

(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.

?

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)}.

?

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 $\mathbb {C}^{2}$ .

?

(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.

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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 $\overrightarrow {v} \otimes \overrightarrow {w}$ .

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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 $\mu =\lambda$ .

?

(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)

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(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}.

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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.

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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}).

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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}.

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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.

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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.

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(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).

?

(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 $n=3$ .

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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 $\Delta (A^{n})$ ?

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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 $A \otimes B + B \otimes A$ ?

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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 $\mathbb {C}^{2}$ ?

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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 $K$ ?

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Problem 5.2.62

(i)(ii)

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

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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).

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Problem 5.2.65

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".

?

(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}.

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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?

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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 $A$ .

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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 $\overrightarrow {v}$ .

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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

?

(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}.

?