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Chapter 1

Basic Operations

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1.1 · Chapter Problems

1.2 · Supplementary Problems

Section 1.1Chapter Problems[81/100%]

Problem 1.1.1

(i)(ii)

Let $\overrightarrow {e}_{1}, \overrightarrow {e}_{2}, \overrightarrow {e}_{3}$ be the standard basis in $\mathbb {R}^{3}$

\overrightarrow {e}_{1} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad \overrightarrow {e}_{2} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \quad \overrightarrow {e}_{3} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}.

Origin: Stated according to the original source

(i)

Consider the normalized vectors

\overrightarrow {a} = \frac{1}{\sqrt{3}}(\overrightarrow {e}_{1}+\overrightarrow {e}_{2}+\overrightarrow {e}_{3}), \quad \overrightarrow {b} = \frac{1}{\sqrt{3}}(-\overrightarrow {e}_{1}-\overrightarrow {e}_{2}+\overrightarrow {e}_{3}),

\overrightarrow {c} = \frac{1}{\sqrt{3}}(-\overrightarrow {e}_{1}+\overrightarrow {e}_{2}-\overrightarrow {e}_{3}), \quad \overrightarrow {d} = \frac{1}{\sqrt{3}}(\overrightarrow {e}_{1}-\overrightarrow {e}_{2}-\overrightarrow {e}_{3}).

These vectors are the unit vectors giving the direction of the four bonds of an atom in the diamond lattice. Show that the four vectors are linearly dependent.

(ii)

Find the scalar products $\overrightarrow {a}^{T}\overrightarrow {b}$ , $\overrightarrow {b}^{T}\overrightarrow {c}$ , $\overrightarrow {c}^{T}\overrightarrow {d}$ , $\overrightarrow {d}^{T}\overrightarrow {a}$ . Discuss.

Problem 1.1.2

(i)(ii)

Origin: Stated according to the original source

(i)

Consider the normalized vector $\overrightarrow {v}$ in $\mathbb {R}^{3}$ and the permutation matrix $P$ , respectively

\overrightarrow {v} = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}, \quad P = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}.

Are the three vectors $\overrightarrow {v}$ , $P\overrightarrow {v}$ , $P^{2}\overrightarrow {v}$ linearly independent?

(ii)

Consider the $4 \times 4$ symmetric matrix $A$ and the vector $\overrightarrow {b}$ in $\mathbb {R}^{4}$

A = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}, \quad \overrightarrow {b} = \begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \end{pmatrix}.

Are the vectors $\overrightarrow {b}$ , $A\overrightarrow {b}$ , $A^{2}\overrightarrow {b}$ , $A^{3}\overrightarrow {b}$ in $\mathbb {R}^{4}$ linearly independent? Show that the matrix $A$ is invertible. Look at the column vectors of the matrix $A$

\begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}, \quad \begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \end{pmatrix}, \quad \begin{pmatrix} 0 \\ 1 \\ 0 \\ 1 \end{pmatrix}, \quad \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}.

Find the inverse of $A$ .

Problem 1.1.3

(i)(ii)

Origin: Stated according to the original source

(i)

Consider the Hilbert space $M_{2}(\mathbb {R})$ of the $2 \times 2$ matrices over $\mathbb {R}$ . Show that the matrices

\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \quad \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}, \quad \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}, \quad \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}

are linearly independent.

(ii)

Use the Gram-Schmidt orthonormalization technique to find an orthonormal basis for $M_{2}(\mathbb {R})$ .

Problem 1.1.4

(i)(ii)

Consider the vector space of $2 \times 2$ matrices over $\mathbb {R}$ and the matrices

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

Origin: Stated according to the original source

(i)

Are these four $2 \times 2$ matrices linearly independent?

(ii)

An $n \times n$ matrix $M$ is called normal if $MM^{*} = M^{*}M$ . Which of the four matrices $A_{j}$ ( $j = 1,2,3,4$ ) are normal matrices?

Problem 1.1.5

(i)(ii)

Let $\sigma_{1}, \sigma_{2}, \sigma_{3}$ be the Pauli spin matrices

\sigma _{1} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma _{2} = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma _{3} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

and $\sigma_{0} = I_{2}$ . Consider the vector space of $2 \times 2$ matrices over $\mathbb {C}$ .

Origin: Stated according to the original source

(i)

Show that the $2 \times 2$ matrices

\frac{1}{\sqrt{2}}I_{2}, \quad \frac{1}{\sqrt{2}}\sigma _{1}, \quad \frac{1}{\sqrt{2}}\sigma _{2}, \quad \frac{1}{\sqrt{2}}\sigma _{3}

are linearly independent.

(ii)

Show that any $2 \times 2$ complex matrix has a unique representation of the form

a_{0}I_{2}+ia_{1}\sigma _{1}+ia_{2}\sigma _{2}+ia_{3}\sigma _{3}

for some $a_{0}, a_{1}, a_{2}, a_{3} \in \mathbb {C}$ .

Problem 1.1.6

Consider the normalized vector $\overrightarrow {v}_{0} = \begin{pmatrix} 1 & 0 & 0 \end{pmatrix}^{T}$ in $\mathbb {R}^{3}$ . Find three normalized vectors $\overrightarrow {v}_{1}, \overrightarrow {v}_{2}, \overrightarrow {v}_{3}$ such that

\sum _{j=0}^{3} \overrightarrow {v}_{j} = \overrightarrow {0}, \quad \overrightarrow {v}_{j}^{T}\overrightarrow {v}_{k} = -\frac{1}{3} \quad (j \neq k).

Origin: Stated according to the original source

Problem 1.1.7

(i)(ii)

Origin: Stated according to the original source

(i)

Find four normalized vectors $\overrightarrow {v}_{1}, \overrightarrow {v}_{2}, \overrightarrow {v}_{3}, \overrightarrow {v}_{4}$ in $\mathbb {R}^{3}$ such that

\overrightarrow {v}_{j}^{T}\overrightarrow {v}_{k} = \frac{4}{3}\delta _{jk}-\frac{1}{3} = \begin{cases} 1 & \text{for } j = k \\ -1/3 & \text{for } j \neq k \end{cases}.

(ii)

Calculate the vector and the matrix, respectively

\sum _{j=1}^{4} \overrightarrow {v}_{j}, \quad \frac{3}{4}\sum _{j=1}^{4} \overrightarrow {v}_{j}\overrightarrow {v}_{j}^{T}.

Problem 1.1.8

Find the set of all four (column) vectors $\overrightarrow {u}_{1}, \overrightarrow {u}_{2}, \overrightarrow {v}_{1}, \overrightarrow {v}_{2}$ in $\mathbb {R}^{2}$ such that the following conditions are satisfied $\overrightarrow {v}_{1}^{T}\overrightarrow {u}_{2} = 0$ , $\overrightarrow {v}_{2}^{T}\overrightarrow {u}_{1} = 0$ , $\overrightarrow {v}_{1}^{T}\overrightarrow {u}_{1} = 1$ , $\overrightarrow {v}_{2}^{T}\overrightarrow {u}_{2} = 1$ .

Origin: Stated according to the original source

Problem 1.1.9

Consider the $2 \times 2$ matrices

A = \begin{pmatrix} a_{11} & a_{12} \\ a_{12} & a_{11} \end{pmatrix}, \quad C = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

where $a_{11}, a_{12} \in \mathbb {R}$ . Can the expression $A^{3}+3AC(A+C)+C^{3}$ be simplified for computation?

Origin: Stated according to the original source

Problem 1.1.10

Let $A$ , $B$ be $2 \times 2$ matrices. Let $AB = 0_{2}$ and $BA = 0_{2}$ . Can we conclude that at least one of the two matrices is the $2 \times 2$ zero matrix? Prove or disprove.

Origin: Stated according to the original source

Problem 1.1.11

Let $A$ , $C$ be $n \times n$ matrices over $\mathbb {R}$ . Let $\overrightarrow {x}$ , $\overrightarrow {y}$ , $\overrightarrow {b}$ , $\overrightarrow {d}$ be column vectors in $\mathbb {R}^{n}$ . Write the system of equations $(A+iC)(\overrightarrow {x}+i\overrightarrow {y}) = (\overrightarrow {b}+i\overrightarrow {d})$ as a $2n \times 2n$ set of real equations.

Origin: Stated according to the original source

Problem 1.1.12

Let $A$ , $B$ be $n \times n$ symmetric matrices over $\mathbb {R}$ , i.e. $A^{T} = A$ , $B^{T} = B$ . What is the condition on $A$ , $B$ such that $AB$ is symmetric?

Origin: Stated according to the original source

Problem 1.1.13

(i)(ii)

Origin: Stated according to the original source

(i)

Compute the matrix product

\begin{pmatrix} x_{1} & x_{2} & x_{3} \end{pmatrix} \begin{pmatrix} 4 & -1 & 2 \\ -1 & 0 & 1 \\ 2 & 1 & 0 \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \end{pmatrix}.

(ii)

Write the quadratic polynomial $3x_{1}^{2}-8x_{1}x_{2}+2x_{2}^{2}+6x_{1}x_{3}-3x_{3}^{2}$ in matrix form.

Problem 1.1.14

An $n \times n$ matrix $M$ is called normal if $MM^{*} = M^{*}M$ . Let $A$ , $B$ be normal $n \times n$ matrices. Assume that $AB^{*} = B^{*}A$ , $BA^{*} = A^{*}B$ . Show that their sum $A+B$ is normal. Show that their product $AB$ is normal.

Origin: Stated according to the original source

Problem 1.1.15

Let $A$ be an $n \times n$ normal matrix. Show that $\operatorname {ker}(A) = \operatorname {ker}(A^{*})$ , where $\operatorname {ker}$ denotes the kernel.

Origin: Stated according to the original source

Problem 1.1.16

(i)(ii)

Origin: Stated according to the original source

(i)

Let $A$ be an $n \times n$ hermitian matrix. Show that $A^{m}$ is a hermitian matrix for all $m \in \mathbb {N}$ .

(ii)

Let $B$ be an $n \times n$ hermitian matrix. Is $iB$ skew-hermitian?

Problem 1.1.17

(i)(ii)

Origin: Stated according to the original source

(i)

Let $A$ be an $n \times n$ matrix with $A^{2} = 0_{n}$ . Is the matrix $I_{n}+A$ invertible?

(ii)

Let $B$ be an $n \times n$ matrix with $B^{3} = 0_{n}$ . Show that $I_{n}+B$ has an inverse.

Problem 1.1.18

Let $A$ be an $n \times n$ matrix over $\mathbb {R}$ . Assume that $A^{2} = 0_{n}$ . Find the inverse of $I_{n}+iA$ .

Origin: Stated according to the original source

Problem 1.1.19

Let $A$ , $B$ be $n \times n$ matrices and $c$ a constant. Assume that the inverses of $(A-cI_{n})$ and $(A+B-cI_{n})$ exist. Show that

(A-cI_{n})^{-1}B(A+B-cI_{n})^{-1} \equiv (A-cI_{n})^{-1}-(A+B-cI_{n})^{-1}.

Origin: Stated according to the original source

Problem 1.1.20

(i)(ii)

Origin: Stated according to the original source

(i)

A $3 \times 3$ matrix $M$ over $\mathbb {R}$ is orthogonal if and only if the columns of $M$ form an orthonormal basis in $\mathbb {R}^{3}$ . Show that the matrix

\begin{pmatrix} \sqrt{3}/3 & 0 & -\sqrt{6}/3 \\ \sqrt{3}/3 & \sqrt{2}/2 & \sqrt{6}/6 \\ \sqrt{3}/3 & -\sqrt{2}/2 & \sqrt{6}/6 \end{pmatrix}

is orthogonal.

(ii)

Represent the nonnormal $3 \times 3$ matrix

A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 2 & 0 \\ -1 & 0 & -1 \end{pmatrix} \quad \text{(relative to the natural basis)}

relative to the orthonormal basis in $\mathbb {R}^{3}$

\left\{ \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \; \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \; \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}\right\} .

Problem 1.1.21

Consider the rotation matrix

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

Let $n$ be a positive integer. Calculate $R^{n}(\theta )$ .

Origin: Stated according to the original source

Problem 1.1.22

(i)(ii)

Origin: Stated according to the original source

(i)

Consider the $2 \times 2$ matrix

A = \begin{pmatrix} 1 & \alpha \\ 1 & \beta \end{pmatrix}

where $\alpha , \beta \in \mathbb {R}$ . Find the condition on $\alpha$ , $\beta$ such that the inverse matrix exists. Find the inverse in this case.

(ii)

Let $t \neq 0$ and $s \in \mathbb {R}$ . Find the inverse of the matrix

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

Problem 1.1.23

Let

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

Can one find a $4 \times 4$ permutation matrix such that $A = PBP^{T}$ ?

Origin: Stated according to the original source

Problem 1.1.24

(i)(ii)

Origin: Stated according to the original source

(i)

Consider the $3 \times 3$ permutation matrix

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

Find all $3 \times 3$ matrices $A$ such that $PAP^{T} = A$ .

(ii)

Consider the $4 \times 4$ permutation matrix

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

Find all $4 \times 4$ matrices $A$ such that $PAP^{T} = A$ .

Problem 1.1.25

(i)(ii)

Origin: Stated according to the original source

(i)

Let $\pi$ be a permutation on $\left\{ 1,2,\ldots ,n\right\}$ . The matrix $P_{\pi }$ for which $p_{i*} = e_{\pi (i)*}$ is called the permutation matrix associated with $\pi$ , where $p_{i*}$ is the $i$ -th row of $P_{\pi }$ and $e_{ij} = 1$ if $i = j$ and 0 otherwise. Let $n = 4$ and $\pi = (3\; 2\; 4\; 1)$ . Find $P_{\pi }$ .

(ii)

Find the corresponding permutation matrix for the permutation

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

Problem 1.1.26

(i)(ii)

Origin: Stated according to the original source

(i)

Find an invertible $2n \times 2n$ matrix $T$ such that

T^{-1} \begin{pmatrix} 0_{n} & I_{n} \\ -I_{n} & 0_{n} \end{pmatrix} T = \begin{pmatrix} 0_{n} & -I_{n} \\ I_{n} & 0_{n} \end{pmatrix}.

(ii)

Show that the $2n \times 2n$ matrix

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

is invertible. Find the inverse.

Problem 1.1.27

Let $M$ be an $2n \times 2n$ matrix with $n \geq 1$ . Then $M$ can be written in block form

M = \begin{pmatrix} A & B \\ C & D \end{pmatrix}

where $A, B, C, D$ are $n \times n$ matrices. Assume that $M^{-1}$ exists and that the $n \times n$ matrix $D$ is also nonsingular. Find $M^{-1}$ using this condition.

Origin: Stated according to the original source

Problem 1.1.28

(i)(ii)

Origin: Stated according to the original source

(i)

Let $A$ be an $m \times n$ matrix with $m \geq n$ . Assume that $A$ has rank $n$ . Show that there exists an $m \times n$ matrix $B$ such that the $n \times n$ matrix $B^{*}A$ is nonsingular. The matrix $B$ can be chosen such that $B^{*}A = I_{n}$ .

(ii)

An $n^{2} \times n$ matrix $J$ is called a selection matrix such that $J^{T}$ is the $n \times n^{2}$ matrix

[E_{11} \; E_{22} \; \ldots \; E_{nn}]

where $E_{ii}$ is the $n \times n$ matrix of zeros except for a 1 in the $(i,i)$ -th position. Find $J$ for $n = 2$ and calculate $J^{T}J$ . Calculate $J^{T}J$ for arbitrary $n$ .

Problem 1.1.29

Let $A$ be an arbitrary $n \times n$ matrix over $\mathbb {R}$ . Can we conclude that $A^{2}$ is positive semidefinite?

Origin: Stated according to the original source

Problem 1.1.30

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

An $n \times n$ matrix $\Pi$ is a projection matrix if $\Pi^{*} = \Pi$ , $\Pi^{2} = \Pi$ .

Origin: Stated according to the original source

(i)

Let $\Pi_{1}$ and $\Pi_{2}$ be projection matrices. Is $\Pi_{1}+\Pi_{2}$ a projection matrix?

(ii)

Let $\Pi_{1}$ and $\Pi_{2}$ be projection matrices. Is $\Pi_{1}\Pi_{2}$ a projection matrix?

(iii)

Let $\Pi$ be a projection matrix. Is $I_{n}-\Pi$ a projection matrix? Calculate $\Pi (I_{n}-\Pi )$ .

(iv)

\Pi = \frac{1}{3} \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}

a projection matrix?

(v)

Let $\epsilon \in [0,1]$ . Show that the $2 \times 2$ matrix

\Pi (\epsilon ) = \begin{pmatrix} \epsilon & \sqrt{\epsilon -\epsilon ^{2}} \\ \sqrt{\epsilon -\epsilon ^{2}} & 1-\epsilon \end{pmatrix}

is a projection matrix. What are the eigenvalues of $\Pi (\epsilon )$ ? Be clever.

Problem 1.1.31

(i)(ii)

Let $m \geq 1$ and $N \geq 2$ . Assume that $N > m$ . Let $X$ be an $N \times m$ matrix over $\mathbb {R}$ such that $X^{*}X = I_{m}$ .

Origin: Stated according to the original source

(i)

We define $\Pi := XX^{*}$ . Calculate $\Pi^{2}$ , $\Pi^{*}$ and $\operatorname {tr}(\Pi )$ .

(ii)

Give an example for such a matrix $X$ , where $m = 1$ and $N = 2$ .

Problem 1.1.32

(i)(ii)

Origin: Stated according to the original source

(i)

A square matrix $A$ is called idempotent if $A^{2} = A$ . Find all $2 \times 2$ matrices $A$ over the real numbers which are idempotent and $a_{ij} \neq 0$ for $i,j = 1,2$ .

(ii)

Let $A$ , $B$ be $n \times n$ idempotent matrices. Show that $A+B$ are idempotent if and only if $AB = BA = 0_{n}$ .

Problem 1.1.33

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Assume that $A+B$ is invertible. Show that

(A+B)^{-1}A = I_{n}-(A+B)^{-1}B, \quad A(A+B)^{-1} = I_{n}-B(A+B)^{-1}.

Origin: Stated according to the original source

Problem 1.1.34

(i)(ii)

Origin: Stated according to the original source

(i)

Find $2 \times 2$ matrices $A$ over $\mathbb {C}$ such that $A^{2} = -I_{2}$ , $A^{*} = -A$ .

(ii)

Find $2 \times 2$ matrices $C$ , $D$ such that $CD = 0_{2}$ and $DC \neq 0_{2}$ .

Problem 1.1.35

Find the $2 \times 2$ matrices $F$ and $F'$ from the two equations

\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} = \frac{1}{2}(F+iF'), \quad \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = \frac{1}{2}(F-iF').

Find the anticommutator of $F$ and $F'$ , i.e. $[F,F']_{+} \equiv FF'+F'F$ .

Origin: Stated according to the original source

Problem 1.1.36

(i)(ii)

Origin: Stated according to the original source

(i)

Find all $2 \times 2$ matrices $X$ such that

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

(ii)

Impose the additional conditions such that $\operatorname {tr}(X) = 0$ and $\operatorname {det}\left(X\right) = +1$ .

Problem 1.1.37

(i)(ii)

Let $\overrightarrow {v}$ be a normalized (column) vector in $\mathbb {C}^{n}$ . Consider the $n \times n$ matrix

A = \overrightarrow {v}\overrightarrow {v}^{*}-\frac{1}{2}I_{n}.

Origin: Stated according to the original source

(i)

Find $A^{*}$ and $AA^{*}$ .

(ii)

Is the matrix $A$ invertible?

Problem 1.1.38

Find all $2 \times 2$ matrices

A = \begin{pmatrix} a_{11} & a_{12} \\ 0 & a_{22} \end{pmatrix}

over $\mathbb {R}$ with $a_{12} \neq 0$ such that $A^{2} = I_{2}$ .

Origin: Stated according to the original source

Problem 1.1.39

(i)(ii)(iii)

Let $z = x+iy$ with $x,y \in \mathbb {R}$ and thus $\bar{z} = x-iy$ . Consider the maps

x+iy \leftrightarrow \begin{pmatrix} x & -y \\ y & x \end{pmatrix} = A(x,y), \quad 1 \leftrightarrow \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad i \leftrightarrow \begin{pmatrix} 0 \\ 1 \end{pmatrix}.

Origin: Stated according to the original source

(i)

Calculate $z^{2}$ and $A^{2}(x,y)$ . Discuss.

(ii)

Find the eigenvalues and normalized eigenvectors of $A(x,y)$ .

(iii)

Calculate $A(x,y) \otimes A(x,y)$ and find the eigenvalues and normalized eigenvectors.

Problem 1.1.40

(i)(ii)

Let $A \in \mathbb {C}^{n \times m}$ with $n \geq m$ and $\operatorname {rank}(A) = m$ .

Origin: Stated according to the original source

(i)

Show that the matrix $m \times m$ matrix $A^{*}A$ is invertible.

(ii)

We set $\Pi = A(A^{*}A)^{-1}A$ . Show that $\Pi$ is a projection matrix, i.e. $\Pi^{2} = \Pi$ and $\Pi = \Pi^{*}$ .

Problem 1.1.41

Let $A$ be an $n \times n$ hermitian matrix and $\Pi$ be an $n \times n$ projection matrix. Then $\Pi A \Pi$ is again a hermitian matrix. Is this still true if $A$ is a normal matrix, i.e. $AA^{*} = A^{*}A$ ?

Origin: Stated according to the original source

Problem 1.1.42

An $n \times n$ matrix $C$ of the form

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

is called a circulant matrix. Show that the set of all $n \times n$ circulant matrices form an $n$ -dimensional vector space.

Origin: Stated according to the original source

Problem 1.1.43

(i)(ii)(iii)

Two $n \times n$ matrices $A$ , $B$ are called similar if there exists an invertible $n \times n$ matrix $S$ such that $A = SBS^{-1}$ . If $A$ is similar to $B$ , then $B$ is also similar to $A$ , since $B = S^{-1}AS$ .

Origin: Stated according to the original source

(i)

Show that the two $2 \times 2$ matrices

A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} = A^{T}

are similar.

(ii)

Consider the two $2 \times 2$ invertible matrices

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

Are the matrices similar?

(iii)

Consider the two $2 \times 2$ invertible matrices

C = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad D = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.

Are the matrices similar?

Problem 1.1.44

(i)(ii)(iii)

An $n \times n$ matrix is called nilpotent if some power of it is the zero matrix, i.e. there is a positive integer $p$ such that $A^{p} = 0_{n}$ .

Origin: Stated according to the original source

(i)

Show that every nonzero nilpotent matrix is nondiagonalizable.

(ii)

Consider the $4 \times 4$ matrix

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

Show that the matrix is nilpotent.

(iii)

Is the matrix product of two $n \times n$ nilpotent matrices nilpotent?

Problem 1.1.45

A matrix $A$ for which $A^{p} = 0_{n}$ , where $p$ is a positive integer, is called nilpotent. If $p$ is the least positive integer for which $A^{p} = 0_{n}$ then $A$ is said to be nilpotent of index $p$ . Find all $2 \times 2$ matrices $A$ over the real numbers which are nilpotent with $p = 2$ , i.e. $A^{2} = 0_{2}$ .

Origin: Stated according to the original source

Problem 1.1.46

(i)(ii)

Let

\overrightarrow {a} = \begin{pmatrix} a_{1} \\ a_{2} \\ a_{3} \end{pmatrix}, \quad \overrightarrow {b} = \begin{pmatrix} b_{1} \\ b_{2} \\ b_{3} \end{pmatrix}

be vectors in $\mathbb {R}^{3}$ . The vector product $\times$ is defined as

\overrightarrow {a} \times \overrightarrow {b} := \begin{pmatrix} a_{2}b_{3}-a_{3}b_{2} \\ a_{3}b_{1}-a_{1}b_{3} \\ a_{1}b_{2}-a_{2}b_{1} \end{pmatrix}.

Origin: Stated according to the original source

(i)

Show that we can find a $3 \times 3$ matrix $S(\overrightarrow {a})$ such that $\overrightarrow {a} \times \overrightarrow {b} = S(\overrightarrow {a})\overrightarrow {b}$ .

(ii)

Express the Jacobi identity

\overrightarrow {a} \times (\overrightarrow {b} \times \overrightarrow {c})+\overrightarrow {c} \times (\overrightarrow {a} \times \overrightarrow {b})+\overrightarrow {b} \times (\overrightarrow {c} \times \overrightarrow {a}) = \overrightarrow {0}

using the matrices $S(\overrightarrow {a})$ , $S(\overrightarrow {b})$ and $S(\overrightarrow {c})$ .

Problem 1.1.47

Let

\overrightarrow {x} = \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \end{pmatrix}, \quad \overrightarrow {y} = \begin{pmatrix} y_{1} \\ y_{2} \\ y_{3} \end{pmatrix}

be two normalized vectors in $\mathbb {R}^{3}$ . Assume that $\overrightarrow {x}^{T}\overrightarrow {y} = 0$ , i.e. the vectors are orthogonal. Is the vector $\overrightarrow {x} \times \overrightarrow {y}$ a unit vector again? We have

x_{1}^{2}+x_{2}^{2}+x_{3}^{2} = 1, \quad y_{1}^{2}+y_{2}^{2}+y_{3}^{2} = 1 \tag {1}

and

\overrightarrow {x}^{T}\overrightarrow {y} = x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3} = 0. \tag {2}

Origin: Stated according to the original source

Problem 1.1.48

(i)(ii)(iii)

Consider the three linear independent normalized column vectors in $\mathbb {R}^{3}$

\overrightarrow {a}_{1} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \quad \overrightarrow {a}_{2} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \quad \overrightarrow {a}_{3} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}.

Origin: Stated according to the original source

(i)

Find the ``volume'' $V_{\overrightarrow {a}} := \overrightarrow {a}_{1}^{T}(\overrightarrow {a}_{2} \times \overrightarrow {a}_{3})$ .

(ii)

From the three vectors $\overrightarrow {a}_{1}$ , $\overrightarrow {a}_{2}$ , $\overrightarrow {a}_{3}$ we form the $3 \times 3$ orthogonal matrix

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

Find the determinant and trace. Discuss.

(iii)

Find the vectors

\overrightarrow {b}_{1} = \frac{1}{V_{\overrightarrow {a}}}\overrightarrow {a}_{2} \times \overrightarrow {a}_{3}, \quad \overrightarrow {b}_{2} = \frac{1}{V_{\overrightarrow {a}}}\overrightarrow {a}_{3} \times \overrightarrow {a}_{1}, \quad \overrightarrow {b}_{3} = \frac{1}{V_{\overrightarrow {a}}}\overrightarrow {a}_{1} \times \overrightarrow {a}_{2}.

Are the three vectors linearly independent?

Problem 1.1.49

Let $\overrightarrow {v}_{1}, \overrightarrow {v}_{2}, \overrightarrow {v}_{3}$ be three normalized linearly independent vectors in $\mathbb {R}^{3}$ . Give an interpretation of $A$ defined by

\cos \left(\frac{1}{2}A\right) = \frac{1+\overrightarrow {v}_{1} \cdot \overrightarrow {v}_{2}+\overrightarrow {v}_{2} \cdot \overrightarrow {v}_{3}+\overrightarrow {v}_{3} \cdot \overrightarrow {v}_{1}}{\sqrt{2(1+\overrightarrow {v}_{1} \cdot \overrightarrow {v}_{2})(1+\overrightarrow {v}_{2} \cdot \overrightarrow {v}_{3})(1+\overrightarrow {v}_{3} \cdot \overrightarrow {v}_{1})}}

where $\cdot$ denotes the scalar product. Consider first the case where $\overrightarrow {v}_{1}, \overrightarrow {v}_{2}, \overrightarrow {v}_{3}$ denote the standard basis in $\mathbb {R}^{3}$ .

Origin: Stated according to the original source

Problem 1.1.50

Let $\overrightarrow {v}$ be a column vector in $\mathbb {R}^{n}$ and $\overrightarrow {v} \neq \overrightarrow {0}$ . Let

A := \frac{\overrightarrow {v}\overrightarrow {v}^{T}}{\overrightarrow {v}^{T}\overrightarrow {v}}

where $^{T}$ denotes the transpose, i.e. $\overrightarrow {v}^{T}$ is a row vector. Calculate $A^{2}$ .

Origin: Stated according to the original source

Problem 1.1.51

(i)(ii)

Origin: Stated according to the original source

(i)

A square matrix $A$ over $\mathbb {C}$ is called skew-hermitian if $A = -A^{*}$ . Show that such a matrix is normal, i.e. we have $AA^{*} = A^{*}A$ .

(ii)

Let $A$ be an $n \times n$ skew-hermitian matrix over $\mathbb {C}$ , i.e. $A^{*} = -A$ . Let $U$ be an $n \times n$ unitary matrix, i.e. $U^{*} = U^{-1}$ . Show that $B := U^{*}AU$ is a skew-hermitian matrix.

Problem 1.1.52

Let $A$ , $X$ , $Y$ be $n \times n$ matrices. Assume that $XA = I_{n}$ , $AY = I_{n}$ . Show that $X = Y$ .

Origin: Stated according to the original source

Problem 1.1.53

(i)(ii)

Origin: Stated according to the original source

(i)

Let $A$ , $B$ be $n \times n$ matrices. Assume that $A$ is nonsingular, i.e. $A^{-1}$ exists. Show that if $BA = 0_{n}$ , then $B = 0_{n}$ .

(ii)

Let $A$ , $B$ be $n \times n$ matrices and $A+B = I_{n}$ , $AB = 0_{n}$ . Show that $A^{2} = A$ and $B^{2} = B$ .

Problem 1.1.54

Find a $2 \times 2$ matrix $A$ over $\mathbb {R}$ such that

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

Origin: Stated according to the original source

Problem 1.1.55

Consider the vector space $\mathbb {R}^{4}$ . Find all pairwise orthogonal vectors (column vectors) $\overrightarrow {x}_{1}, \ldots , \overrightarrow {x}_{p}$ , where the entries of the column vectors can only be $+1$ or $-1$ . Calculate the matrix

\sum _{j=1}^{p} \overrightarrow {x}_{j}\overrightarrow {x}_{j}^{T}

and find the eigenvalues and eigenvectors of this matrix.

Origin: Stated according to the original source

Problem 1.1.56

The $(n+1) \times (n+1)$ Hadamard matrix $H(n)$ of any dimension is generated recursively as follows

H(n) = \begin{pmatrix} H(n-1) & H(n-1) \\ H(n-1) & -H(n-1) \end{pmatrix}

where $n = 1,2,\ldots$ and $H(0)$ is the $1 \times 1$ matrix $H(0) = (1)$ . The column vectors in the matrix $H(n)$ form a basis in $\mathbb {R}^{n}$ . They are pairwise orthogonal to each other. Find $H(1)$ , $H(2)$ , $H(3)$ and $H(3)H^{T}(3)$ . Find the eigenvalues of $H(1)$ and $H(2)$ .

Origin: Stated according to the original source

Problem 1.1.57

A Hadamard matrix is an $n \times n$ matrix $H$ with entries in $\left\{ -1,+1\right\}$ such that any two distinct rows or columns of $H$ have inner product 0. Construct a $4 \times 4$ Hadamard matrix starting from the column vector $\overrightarrow {x}_{1} = \begin{pmatrix} 1 & 1 & 1 & 1 \end{pmatrix}^{T}$ .

Origin: Stated according to the original source

Problem 1.1.58

(i)(ii)(iii)

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

A = \begin{pmatrix} 2 & 2 & -2 \\ 2 & 2 & -2 \\ -2 & -2 & 6 \end{pmatrix}.

Origin: Stated according to the original source

(i)

Let $X$ be an $m \times n$ matrix. The column rank of $X$ is the maximum number of linearly independent columns. The row rank is the maximum number of linearly independent rows. The row rank and the column rank of $X$ are equal (called the rank of $X$ ). Find the rank of $A$ and denote it by $k$ .

(ii)

Locate a $k \times k$ submatrix of $A$ having rank $k$ .

(iii)

Find $3 \times 3$ permutation matrices $P$ and $Q$ such that in the matrix $PAQ$ the submatrix from (ii) is in the upper left portion of $A$ .

Problem 1.1.59

Find all $n \times n$ matrices $A$ over $\mathbb {R}$ that satisfy the equation $A^{T}A = 0_{n}$ .

Origin: Stated according to the original source

Problem 1.1.60

(i)(ii)

A square matrix $A$ such that $A^{2} = I_{n}$ is called involutory.

Origin: Stated according to the original source

(i)

Find all $2 \times 2$ matrices over the real numbers which are involutory. Assume that $a_{ij} \neq 0$ for $i,j = 1,2$ .

(ii)

Show that an $n \times n$ matrix $A$ is involutory if and only if $(I_{n}-A)(I_{n}+A) = 0_{n}$ .

Problem 1.1.61

(i)(ii)

Origin: Stated according to the original source

(i)

Let $A$ be an $n \times n$ symmetric matrix over $\mathbb {R}$ . Let $P$ be an arbitrary $n \times n$ matrix over $\mathbb {R}$ . Show that $P^{T}AP$ is symmetric.

(ii)

Let $B$ be an $n \times n$ skew-symmetric matrix over $\mathbb {R}$ , i.e. $B^{T} = -B$ . Let $P$ be an arbitrary $n \times n$ matrix over $\mathbb {R}$ . Show that $P^{T}BP$ is skew-symmetric.

Problem 1.1.62

Let $A$ be an $m \times n$ matrix. The column rank of $A$ is the maximum number of linearly independent columns. The row rank is the maximum number of linearly independent rows. The row rank and the column rank of $A$ are equal (called the rank of $A$ ). Find the rank of the $4 \times 4$ matrix

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

Origin: Stated according to the original source

Problem 1.1.63

A Cartan matrix $A$ is a square matrix whose elements $a_{ij}$ satisfy the following conditions:

$a_{ij}$ is an integer, one of $\left\{ -3,-2,-1,0,2\right\}$
$a_{jj} = 2$ for all diagonal elements of $A$
$a_{ij} \leq 0$ off of the diagonal
$a_{ij} = 0$ iff $a_{ji} = 0$
There exists an invertible diagonal matrix $D$ such that $DAD^{-1}$ gives a symmetric and positive definite quadratic form. Give a $2 \times 2$ non-diagonal Cartan matrix and its eigenvalues.

Origin: Stated according to the original source

Problem 1.1.64

Let $A$ , $B$ , $C$ , $D$ be $n \times n$ matrices over $\mathbb {R}$ . Assume that $AB^{T}$ and $CD^{T}$ are symmetric and $AD^{T}-BC^{T} = I_{n}$ , where $^{T}$ denotes transpose. Show that $A^{T}D-C^{T}B = I_{n}$ .

Origin: Stated according to the original source

Problem 1.1.65

Let $n$ be a positive integer. Let $A_{n}$ be the $(2n+1) \times (2n+1)$ skew-symmetric matrix for which each entry in the first $n$ subdiagonals below the main diagonal is 1 and each of the remaining entries below the main diagonal is $-1$ . Give $A_{1}$ and $A_{2}$ . Find the rank of $A_{n}$ .

Origin: Stated according to the original source

Problem 1.1.66

(i)(ii)

A vector $\overrightarrow {u} = (u_{1},u_{2},\ldots ,u_{n})$ is called a probability vector if the components are nonnegative and their sum is 1.

Origin: Stated according to the original source

(i)

Is the vector $\overrightarrow {u} = (1/2,0,1/4,1/4)$ a probability vector?

(ii)

Can the vector $\overrightarrow {v} = (2,3,5,1,0)$ be ``normalized'' so that we obtain a probability vector?

Problem 1.1.67

An $n \times n$ matrix $P = (p_{ij})$ is called a stochastic matrix if each of its rows is a probability vector, i.e. if each entry of $P$ is nonnegative and the sum of the entries in each row is 1. Let $A$ and $B$ be two stochastic $n \times n$ matrices. Is the matrix product $AB$ also a stochastic matrix?

Origin: Stated according to the original source

Problem 1.1.68

(i)(ii)

The numerical range, also known as the field of values, of an $n \times n$ matrix $A$ over the complex numbers, is defined as

F(A) := \left\{ \overrightarrow {z}^{*}A\overrightarrow {z} : \left\| \overrightarrow {z}\right\| = \overrightarrow {z}^{*}\overrightarrow {z} = 1, \; \overrightarrow {z} \in \mathbb {C}^{n}\right\} .

The Toeplitz-Hausdorff convexity theorem tells us that the numerical range of a square matrix is a convex compact subset of the complex plane.

Origin: Stated according to the original source

(i)

Find the numerical range for the $2 \times 2$ matrices

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

(ii)

Let $\alpha \in \mathbb {R}$ and the $5 \times 5$ symmetric matrix

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

Show that the set $F(A)$ lies on the real axis. Show that $\left|\overrightarrow {z}^{*}A\overrightarrow {z}\right| \leq \alpha +8$ .

Problem 1.1.69

(i)(ii)

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ and $F(A)$ the field of values. Let $U$ be an $n \times n$ unitary matrix.

Origin: Stated according to the original source

(i)

Show that $F(U^{*}AU) = F(A)$ .

(ii)

Apply the theorem to the two $2 \times 2$ matrices which are unitarily equivalent

A_{1} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad A_{2} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.

Problem 1.1.70

(i)(ii)

Let $A$ be an $m \times n$ matrix over $\mathbb {C}$ . The Moore-Penrose pseudo inverse matrix $A^{+}$ is the unique $n \times m$ matrix which satisfies

AA^{+}A = A, \quad A^{+}AA^{+} = A^{+}, \quad (AA^{+})^{*} = AA^{+}, \quad (A^{+}A)^{*} = A^{+}A.

We also have that

\overrightarrow {x} = A^{+}\overrightarrow {b} \tag {1}

is the shortest length least square solution to the problem $A\overrightarrow {x} = \overrightarrow {b}$ .

Origin: Stated according to the original source

(i)

Show that if $(A^{*}A)^{-1}$ exists, then $A^{+} = (A^{*}A)^{-1}A^{*}$ .

(ii)

Let

A = \begin{pmatrix} 1 & 3 \\ 2 & 4 \\ 3 & 5 \end{pmatrix}.

Find the Moore-Penrose matrix inverse $A^{+}$ of $A$ .

Problem 1.1.71

The Fibonacci numbers are defined by the recurrence relation (linear difference equation of second order with constant coefficients) $s_{n+2} = s_{n+1}+s_{n}$ , where $n = 0,1,\ldots$ and $s_{0} = 0$ , $s_{1} = 1$ . Write this recurrence relation in matrix form. Find $s_{6}$ , $s_{5}$ , and $s_{4}$ .

Origin: Stated according to the original source

Problem 1.1.72

Assume that $A = A_{1}+iA_{2}$ is a nonsingular $n \times n$ matrix, where $A_{1}$ and $A_{2}$ are real $n \times n$ matrices. Assume that $A_{1}$ is also nonsingular. Find the inverse of $A$ using the inverse of $A_{1}$ .

Origin: Stated according to the original source

Problem 1.1.73

Let $A$ and $B$ be $n \times n$ matrices over $\mathbb {R}$ . Assume that $A \neq B$ , $A^{3} = B^{3}$ and $A^{2}B = B^{2}A$ . Is $A^{2}+B^{2}$ invertible?

Origin: Stated according to the original source

Problem 1.1.74

Let $A$ be a symmetric $2 \times 2$ matrix over $\mathbb {R}$

A = \begin{pmatrix} a_{00} & a_{01} \\ a_{10} & a_{11} \end{pmatrix}.

Thus $a_{01} = a_{10}$ . Assume that $a_{00}a_{01} = a_{01}^{2}$ , $a_{00}a_{11} = a_{01}a_{11}$ . Find all matrices $A$ that satisfy these conditions.

Origin: Stated according to the original source

Problem 1.1.75

Consider the invertible $2 \times 2$ matrix

J = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \Rightarrow J^{-1} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.

Find all nonzero $2 \times 2$ matrices $A$ such that $AJ = JA$ .

Origin: Stated according to the original source

Problem 1.1.76

(i)(ii)(iii)

Consider the column vector in $\mathbb {R}^{8}$

\overrightarrow {x} = (20.0 \; 6.0 \; 4.0 \; 2.0 \; 10.0 \; 6.0 \; 8.0 \; 4.0)^{T}

where $^{T}$ denotes transpose. Consider the matrices

H_{1} = \frac{1}{2} \begin{pmatrix} 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{pmatrix}

G_{1} = \frac{1}{2} \begin{pmatrix} 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 \end{pmatrix}

H_{2} = \frac{1}{2} \begin{pmatrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix}, \quad G_{2} = \frac{1}{2} \begin{pmatrix} 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 \end{pmatrix}

H_{3} = \frac{1}{2}\begin{pmatrix} 1 & 1 \end{pmatrix}, \quad G_{3} = \frac{1}{2}\begin{pmatrix} 1 & -1 \end{pmatrix}.

Origin: Stated according to the original source

(i)

Calculate the vectors

H_1x, G_1x H_2H_1x, G_2H_1x, H_2G_1x, G_2G_1x H_3H_2H_1x, G_3H_2H_1x, H_3G_2H_1x, G_3G_2H_1x, H_3H_2G_1x, G_3H_2G_1x, H_3G_2G_1x, G_3G_2G_1x.

(ii)

Calculate $H_{j}H_{j}^{T}$ , $G_{j}G_{j}^{T}$ , $H_{j}G_{j}^{T}$ for $j = 1,2,3$ .

(iii)

How can we reconstruct the original vector $\overrightarrow {x}$ from the vector

(H_{3}H_{2}H_{1}\overrightarrow {x}, G_{3}H_{2}H_{1}\overrightarrow {x}, H_{3}G_{2}H_{1}\overrightarrow {x}, G_{3}G_{2}H_{1}\overrightarrow {x}, H_{3}H_{2}G_{1}\overrightarrow {x}, G_{3}H_{2}G_{1}\overrightarrow {x}, H_{3}G_{2}G_{1}\overrightarrow {x}, G_{3}G_{2}G_{1}\overrightarrow {x}).

The problem plays a role in wavelet theory.

Problem 1.1.77

(i)(ii)

Given a signal as the column vector

\overrightarrow {x} = (3.0 \; 0.5 \; 2.0 \; 7.0)^{T}.

The pyramid algorithm (for Haar wavelets) is as follows: The first two entries $(3.0 \; 0.5)^{T}$ in the signal give an average of $(3.0+0.5)/2 = 1.75$ and a difference average of $(3.0-0.5)/2 = 1.25$ . The second two entries $(2.0 \; 7.0)$ give an average of $(2.0+7.0)/2 = 4.5$ and a difference average of $(2.0-7.0)/2 = -2.5$ . Thus we end up with a vector

(1.75 \; 1.25 \; 4.5 \; -2.5)^{T}.

Now we take the average of 1.75 and 4.5 providing $(1.75+4.5)/2 = 3.125$ and the difference average $(1.75-4.5)/2 = -1.375$ . Thus we end up with the vector

\overrightarrow {y} = (3.125 \; -1.375 \; 1.25 \; -2.5)^{T}.

Origin: Stated according to the original source

(i)

Find a $4 \times 4$ matrix $A$ such that

\overrightarrow {x} \equiv \begin{pmatrix} 3.0 \\ 0.5 \\ 2.0 \\ 7.0 \end{pmatrix} = A\overrightarrow {y} \equiv A \begin{pmatrix} 3.125 \\ -1.375 \\ 1.25 \\ -2.5 \end{pmatrix}.

(ii)

Show that the inverse of $A$ exists. Then find the inverse.

Problem 1.1.78

The $8 \times 8$ matrix

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

plays a role in the discrete wavelet transform. Show that the matrix is invertible without calculating the determinant. Find the inverse.

Origin: Stated according to the original source

Problem 1.1.79

(i)(ii)

Origin: Stated according to the original source

(i)

For $n = 4$ the transform matrix for the Daubechies wavelet is given by

D_{4} = \begin{pmatrix} c_{0} & c_{1} & c_{2} & c_{3} \\ c_{3} & -c_{2} & c_{1} & -c_{0} \\ c_{2} & c_{3} & c_{0} & c_{1} \\ c_{1} & -c_{0} & c_{3} & -c_{2} \end{pmatrix}, \quad \begin{pmatrix} c_{0} \\ c_{1} \\ c_{2} \\ c_{3} \end{pmatrix} = \frac{1}{4\sqrt{2}} \begin{pmatrix} 1+\sqrt{3} \\ 3+\sqrt{3} \\ 3-\sqrt{3} \\ 1-\sqrt{3} \end{pmatrix}.

Is $D_{4}$ orthogonal? Prove or disprove.

(ii)

For $n = 8$ the transform matrix for the Daubechies wavelet is given by

D_{8} = \begin{pmatrix} c_{0} & c_{1} & c_{2} & c_{3} & 0 & 0 & 0 & 0 \\ c_{3} & -c_{2} & c_{1} & -c_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & c_{0} & c_{1} & c_{2} & c_{3} & 0 & 0 \\ 0 & 0 & c_{3} & -c_{2} & c_{1} & -c_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & c_{0} & c_{1} & c_{2} & c_{3} \\ 0 & 0 & 0 & 0 & c_{3} & -c_{2} & c_{1} & -c_{0} \\ c_{2} & c_{3} & 0 & 0 & 0 & 0 & c_{0} & c_{1} \\ c_{1} & -c_{0} & 0 & 0 & 0 & 0 & c_{3} & -c_{2} \end{pmatrix}.

Is $D_{8}$ orthogonal? Prove or disprove.

Problem 1.1.80

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Assume that $A$ and $A+B$ are invertible. Then $(A+B)^{-1} \equiv A^{-1}-A^{-1}B(A+B)^{-1}$ . Apply the identity to $A = \sigma_{1}$ , $B = \sigma_{3}$ .

Origin: Stated according to the original source

Problem 1.1.81

Let $A$ be a positive definite $n \times n$ matrix over $\mathbb {R}$ . Let $\overrightarrow {x} \in \mathbb {R}^{n}$ . Show that $A+\overrightarrow {x}\overrightarrow {x}^{T}$ is positive definite.

Origin: Stated according to the original source

Section 1.2Supplementary Problems[35/0%]

Problem 1.2.1

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

Are the four $2 \times 2$ nonnormal matrices

\frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}, \quad \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 0 \\ 1 & -1 \end{pmatrix}, \quad \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -1 \\ 0 & 0 \end{pmatrix}, \quad \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix}

linearly independent?

(ii)

Show that the nine $3 \times 3$ matrices

\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \quad \begin{pmatrix} 1 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \quad \begin{pmatrix} 1 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \quad \begin{pmatrix} 1 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}, \quad \begin{pmatrix} 1 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \end{pmatrix},

\begin{pmatrix} 1 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix}, \quad \begin{pmatrix} 1 & 1 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \end{pmatrix}, \quad \begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \end{pmatrix}, \quad \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}

form a basis in the vector space of the $3 \times 3$ matrices. This basis is called the spiral basis (and can be extended to any dimension). Which of these matrices are nonnormal?

(iii)

Do the eight $3 \times 3$ skew-hermitian matrices

\Gamma _{1} = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \quad \Gamma _{2} = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix},

\Gamma _{3} = \frac{1}{\sqrt{2}} \begin{pmatrix} i & 0 & 0 \\ 0 & -i & 0 \\ 0 & 0 & 0 \end{pmatrix}, \quad \Gamma _{4} = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 0 & i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{pmatrix},

\Gamma _{5} = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix}, \quad \Gamma _{6} = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & i \\ 0 & i & 0 \end{pmatrix},

\Gamma _{7} = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix}, \quad \Gamma _{8} = \frac{1}{\sqrt{6}} \begin{pmatrix} i & 0 & 0 \\ 0 & i & 0 \\ 0 & 0 & -2i \end{pmatrix}

together with the $3 \times 3$ unit matrix form an orthonormal basis in the vector space of $3 \times 3$ matrices over the complex number. Find all pairwise scalar products $\left\langle A, B \right\rangle := \operatorname {tr}(AB^{*})$ . Discuss.

Problem 1.2.2

(i)(ii)

Origin: Stated according to the original source

(i)

One can describe a tetrahedron in the vector space $\mathbb {R}^{3}$ by specifying vectors $\overrightarrow {v}_{1}, \overrightarrow {v}_{2}, \overrightarrow {v}_{3}, \overrightarrow {v}_{4}$ normal to its faces with lengths equal to the faces' area. Give an example.

(ii)

Consider a tetrahedron defined by the triple of linearly independent vectors $\overrightarrow {v}_{j} \in \mathbb {R}^{3}$ , $j = 1,2,3$ . Show that the normal vectors to the faces defined by two of these vectors, normalized to the area of the face, is given by

\overrightarrow {n}_{1} = \frac{1}{2}\overrightarrow {v}_{2} \times \overrightarrow {v}_{3}, \quad \overrightarrow {n}_{2} = \frac{1}{2}\overrightarrow {v}_{3} \times \overrightarrow {v}_{1}, \quad \overrightarrow {n}_{3} = \frac{1}{2}\overrightarrow {v}_{1} \times \overrightarrow {v}_{2}.

Show that

\overrightarrow {v}_{1} = \frac{2}{3V}\overrightarrow {n}_{2} \times \overrightarrow {n}_{3}, \quad \overrightarrow {v}_{2} = \frac{2}{3V}\overrightarrow {n}_{3} \times \overrightarrow {n}_{1}, \quad \overrightarrow {v}_{3} = \frac{2}{3V}\overrightarrow {n}_{1} \times \overrightarrow {n}_{2}

where $V$ is the volume of the tetrahedron given by

V = \frac{1}{3!}(\overrightarrow {v}_{1} \times \overrightarrow {v}_{2}) \cdot \overrightarrow {v}_{3} = \sqrt{\frac{2}{9}(\overrightarrow {n}_{1} \times \overrightarrow {n}_{2}) \cdot \overrightarrow {n}_{3}}.

Apply it to normalized vectors which form an orthonormal basis in $\mathbb {R}^{3}$

\overrightarrow {n}_{1} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \quad \overrightarrow {n}_{2} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \quad \overrightarrow {n}_{3} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}.

Problem 1.2.3

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

Show that any rank one positive semidefinite $n \times n$ matrix $M$ can be written as $M = \overrightarrow {v}\overrightarrow {v}^{T}$ , where $\overrightarrow {v}$ is some (column) vector in $\mathbb {C}^{n}$ .

(ii)

Let $\overrightarrow {u}$ , $\overrightarrow {v}$ be normalized (column) vectors in $\mathbb {C}^{n}$ . Let $A$ be an $n \times n$ positive semidefinite matrix over $\mathbb {C}$ . Show that $(\overrightarrow {u}^{*}\overrightarrow {v})(\overrightarrow {u}^{*}A\overrightarrow {v}) \geq 0$ .

(iii)

Show that if hermitian matrices $S$ and $T$ are positive semidefinite and commute ( $ST = TS$ ), then their product $ST$ is also positive semidefinite. We have to show that

(ST\overrightarrow {v})^{*}\overrightarrow {v} \geq 0

for all $\overrightarrow {v} \in \mathbb {C}^{n}$ . Note that if $S = 0_{n}$ , then obviously $(STu)^{*}\overrightarrow {u} = 0$ . So assume that $S \neq 0_{n}$ , i.e. $\left\| S\right\| > 0$ .

Problem 1.2.4

Show that the symmetric matrices

\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}, \quad \begin{pmatrix} 1 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 1 \end{pmatrix}, \quad \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \end{pmatrix}

are positive semidefinite. Extend to the $n \times n$ case.

Origin: Stated according to the original source

Problem 1.2.5

Let $A$ be a hermitian $n \times n$ matrix and $A \neq 0_{n}$ . Show that $A^{m} \neq 0_{n}$ for all $m \in \mathbb {N}$ .

Origin: Stated according to the original source

Problem 1.2.6

Let $A \in \mathbb {R}^{m \times n}$ be a nonzero matrix. Let $\overrightarrow {x} \in \mathbb {R}^{n}$ , $\overrightarrow {y} \in \mathbb {R}^{m}$ be vectors such that $c := \overrightarrow {y}^{T}A\overrightarrow {x} \neq 0$ . Show that the matrix $B := A-c^{-1}A\overrightarrow {x}\overrightarrow {y}^{T}A$ has rank exactly one less than the rank of $A$ .

Origin: Stated according to the original source

Problem 1.2.7

(i)(ii)

Origin: Stated according to the original source

(i)

Consider the two-dimensional Euclidean space and let $\overrightarrow {e}_{1}$ , $\overrightarrow {e}_{2}$ be the standard basis. Consider the seven vectors

\overrightarrow {v}_{0} = \overrightarrow {0}, \quad \overrightarrow {v}_{1} = \frac{1}{2}\overrightarrow {e}_{1}+\frac{\sqrt{3}}{2}\overrightarrow {e}_{2}, \quad \overrightarrow {v}_{2} = -\frac{1}{2}\overrightarrow {e}_{1}-\frac{\sqrt{3}}{2}\overrightarrow {e}_{2},

\overrightarrow {v}_{3} = -\frac{1}{2}\overrightarrow {e}_{1}+\frac{\sqrt{3}}{2}\overrightarrow {e}_{2}, \quad \overrightarrow {v}_{4} = \frac{1}{2}\overrightarrow {e}_{1}-\frac{\sqrt{3}}{2}\overrightarrow {e}_{2}, \quad \overrightarrow {v}_{5} = -\overrightarrow {e}_{1}, \quad \overrightarrow {v}_{6} = \overrightarrow {e}_{1}.

Find the distance between the vectors and select the vectors pairs with the shortest distance.

(ii)

Consider the vectors in $\mathbb {R}^{2}$

\overrightarrow {v}_{k} = \begin{pmatrix} \cos ((k-1)\pi /4) \\ \sin ((k-1)\pi /4) \end{pmatrix}, \quad k = 1,3,5,7

and

\overrightarrow {v}_{k} = \sqrt{2} \begin{pmatrix} \cos ((k-1)\pi /4) \\ \sin ((k-1)\pi /4) \end{pmatrix}, \quad k = 2,4,6,8

which play a role for the lattice Boltzmann model. Find the angles between the vectors.

Problem 1.2.8

Show that the $4 \times 4$ matrices

\Pi = \frac{1}{2} \begin{pmatrix} 1 & 1/\sqrt{2} & 0 & -1/\sqrt{2} \\ 1/\sqrt{2} & 1 & 1/\sqrt{2} & 0 \\ 0 & 1/\sqrt{2} & 1 & 1/\sqrt{2} \\ -1/\sqrt{2} & 0 & 1/\sqrt{2} & 1 \end{pmatrix}, \quad \tilde{\Pi } = I_{4}-\Pi

are projection matrices. Describe the subspaces of $\mathbb {R}^{4}$ they project into.

Origin: Stated according to the original source

Problem 1.2.9

(i)(ii)

Origin: Stated according to the original source

(i)

The vectors $\overrightarrow {u}$ , $\overrightarrow {v}$ , $\overrightarrow {w}$ point to the vertices of a equilateral triangle

\overrightarrow {u} = \begin{pmatrix} 1/\sqrt{3} \\ 0 \end{pmatrix}, \quad \overrightarrow {v} = \begin{pmatrix} -1/(2\sqrt{3}) \\ 1/2 \end{pmatrix}, \quad \overrightarrow {w} = \begin{pmatrix} -1/(2\sqrt{3}) \\ -1/2 \end{pmatrix}.

Find the area of this triangle.

(ii)

Let $\overrightarrow {u}$ , $\overrightarrow {v}$ be (column) vectors in $\mathbb {R}^{n}$ . Show that

A = \sqrt{\left|(\overrightarrow {u}^{T}\overrightarrow {u})(\overrightarrow {v}^{T}\overrightarrow {v})-(\overrightarrow {u}^{T}\overrightarrow {v})^{2}\right|}

calculates the area spanned by the vectors $\overrightarrow {u}$ and $\overrightarrow {v}$ .

Problem 1.2.10

Assume that two planes in $\mathbb {R}^{3}$ given by

kx_{1}+\ell x_{2}+mx_{3}+n = 0, \quad k'x_{1}+\ell 'x_{2}+m'x_{3}+n' = 0

be the mirror images with respect to a third plane in $\mathbb {R}^{3}$ given by

ax_{1}+bx_{2}+cx_{3}+d = 0.

Show that ( $r^{2} = a^{2}+b^{2}+c^{2}$ )

\begin{pmatrix} k' \\ \ell ' \\ m' \end{pmatrix} = \frac{1}{r^{2}} \begin{pmatrix} a^{2}-b^{2}-c^{2} & 2ab & 2ac \\ 2ab & -a^{2}+b^{2}-c^{2} & 2bc \\ 2ac & 2bc & -a^{2}-b^{2}+c^{2} \end{pmatrix} \begin{pmatrix} k \\ \ell \\ m \end{pmatrix}.

Origin: Stated according to the original source

Problem 1.2.11

(i)(ii)

Origin: Stated according to the original source

(i)

Let $A$ be a hermitian $n \times n$ matrix over $\mathbb {C}$ with $A^{2} = I_{n}$ . Find the matrix $(A^{-1}+iI_{n})^{-1}$ .

(ii)

Let $B$ be an $n \times n$ matrix over $\mathbb {C}$ . Find the condition on $B$ such that

(I_{n}+iB)(I_{n}-iB) = I_{n}.

This means that $(I_{n}-iB)$ is the inverse of $(I_{n}+iB)$ .

Problem 1.2.12

Consider the vector space $\mathbb {R}^{d}$ . Suppose that $\left\{ \overrightarrow {v}_{j}\right\}_{j=1}^{d}$ and $\left\{ \overrightarrow {w}_{k}\right\}_{k=1}^{d}$ are two bases in $\mathbb {R}^{d}$ . Then there is an invertible $d \times d$ matrix

T = (t_{jk})_{j,k=1}^{d}

so that

\overrightarrow {v}_{j} = \sum _{k=1}^{d} t_{jk}\overrightarrow {w}_{k}, \quad j = 1,\ldots ,d.

The bases $\left\{ \overrightarrow {v}_{j}\right\}_{j=1}^{d}$ and $\left\{ \overrightarrow {w}_{k}\right\}_{k=1}^{d}$ are said to have the same orientation if $\operatorname {det}\left(T\right) > 0$ . If $\operatorname {det}\left(T\right) < 0$ , then they have the opposite orientation. Consider the two bases in $\mathbb {R}^{2}$

\left\{ \overrightarrow {v}_{1} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \; \overrightarrow {v}_{2} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\right\} , \quad \left\{ \overrightarrow {w}_{1} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \; \overrightarrow {w}_{2} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix}\right\} .

Find the orientation.

Origin: Stated according to the original source

Problem 1.2.13

(i)(ii)

Origin: Stated according to the original source

(i)

Let $\omega > 0$ (fixed) and $t \geq 0$ . Show that the $4 \times 4$ matrix

S(t) = \frac{1}{4} \begin{pmatrix} 1+3e^{-\omega t} & 1-e^{-\omega t} & 1-e^{-\omega t} & 1-e^{-\omega t} \\ 1-e^{-\omega t} & 1+3e^{-\omega t} & 1-e^{-\omega t} & 1-e^{-\omega t} \\ 1-e^{-\omega t} & 1-e^{-\omega t} & 1+3e^{-\omega t} & 1-e^{-\omega t} \\ 1-e^{-\omega t} & 1-e^{-\omega t} & 1-e^{-\omega t} & 1+3e^{-\omega t} \end{pmatrix}

is a stochastic matrix (probabilistic matrix). Find the eigenvalues of $S(t)$ . Let

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

Find the vectors $\pi S(t)$ , $(\pi S(t))S(t)$ etc.. Is $S(t_{1}+t_{2}) = S(t_{1})S(t_{2})$ ?

(ii)

Let $\epsilon \in [0,1]$ . Consider the stochastic matrix

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

Let $n = 1,2,\ldots$ . Show that

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

Problem 1.2.14

Can one find $4 \times 4$ matrices $A$ and $B$ such that

A \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix} B = \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \end{pmatrix} \; ?

Of course at least one of the matrices $A$ and $B$ must be singular. The underlying field $\mathbb {F}$ could be $\mathbb {R}$ or $\operatorname {char}(\mathbb {F}) = 2$ .

Origin: Stated according to the original source

Problem 1.2.15

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

Origin: Stated according to the original source

(i)

Find all $2 \times 2$ matrices such that $A^{2} \neq 0_{2}$ but $A^{3} = 0_{2}$ .

(ii)

Find all non-hermitian $2 \times 2$ matrices $B$ such that $BB^{*} = I_{2}$ .

(iii)

Find all hermitian $2 \times 2$ matrices $H$ such that $H^{2} = 0_{2}$ .

(iv)

Find all $2 \times 2$ matrices $C$ over $\mathbb {R}$ such that $C^{T}C+CC^{T} = I_{2}$ , $C^{2} = 0_{2}$ .

(v)

Find all $2 \times 2$ matrices $A_{1}$ , $A_{2}$ , $A_{3}$ such that $A_{1}A_{2} = A_{2}A_{3}$ , $A_{3}A_{1} = A_{2}A_{3}$ .

(vi)

Find all $2 \times 2$ matrices $A$ and $B$ such that $A^{2} = B^{2} = I_{2}$ , $AB = -I_{2}$ .

Problem 1.2.16

Let $A$ be an $m \times n$ matrix and $B$ be a $p \times q$ matrix. Then the direct sum of $A$ and $B$ , denoted by $A \oplus B$ , is the $(m+p) \times (n+q)$ matrix defined by

A \oplus B := \begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix}.

Let $A_{1}$ , $A_{2}$ be $m \times m$ matrices and $B_{1}$ , $B_{2}$ be $n \times n$ matrices. Show that

(A_{1} \oplus B_{1})(A_{2} \oplus B_{2}) = (A_{1}A_{2}) \oplus (B_{1}B_{2}).

Origin: Stated according to the original source

Problem 1.2.17

Consider the symmetric matrix $A$ and the normalized vector $\overrightarrow {v}$

A = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}, \quad \overrightarrow {v} = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}.

Calculate $S_{j} = \overrightarrow {v}^{*}A^{j}\overrightarrow {v}$ for $j = 1,2,3$ . Can $A$ be reconstructed from $S_{1}, S_{2}, S_{3}$ and the information that $A$ is real symmetric?

Origin: Stated according to the original source

Problem 1.2.18

(i)(ii)

The standard simplex $\Delta_{n}$ is defined by the set in $\mathbb {R}^{n}$

\Delta _{n} := \left\{ (x_{1},\ldots ,x_{n})^{T} : x_{j} \geq 0, \; \sum _{j=1}^{n} x_{j} = 1\right\} .

Consider $n$ affinely independent points $B_{1},\ldots ,B_{n} \in \Delta_{n}$ . They span an $(n-1)$ -simplex denoted by $\Lambda = \operatorname {Con}(B_{1},\ldots ,B_{n})$ , that is

\Lambda = \operatorname {Con}(B_{1},\ldots ,B_{n}) = \left\{ \lambda _{1}B_{1}+\cdots +\lambda _{n}B_{n} : \sum _{j=1}^{n} \lambda _{j} = 1, \; \lambda _{1},\ldots ,\lambda _{n} \geq 0\right\} .

The set corresponds to an invertible $n \times n$ matrix $[\Lambda ]$ whose columns are $B_{1},\ldots ,B_{n}$ . Conversely, consider the matrix $C = (b_{jk})$ , where $C_{k} = (b_{1k},\ldots ,b_{nk})^{T}$ ( $k = 1,\ldots ,n$ ). If $\operatorname {det}\left(C\right) \neq 0$ and the sum of the entries in each column is 1, then the matrix $C$ corresponds to an $(n-1)$ -simplex $\operatorname {Con}(B_{1},\ldots ,B_{n})$ in $\Delta_{n}$ . Let $C_{1}$ and $C_{2}$ be $n \times n$ matrices with nonnegative entries and all the columns of each matrix add up to 1.

Origin: Stated according to the original source

(i)

Show that $C_{1}C_{2}$ and $C_{2}C_{1}$ are also such matrices.

(ii)

Are the $n^{2} \times n^{2}$ matrices $C_{1} \otimes C_{2}$ , $C_{2} \otimes C_{1}$ such matrices?

Problem 1.2.19

(i)(ii)

Origin: Stated according to the original source

(i)

Let $S$ , $T$ be $n \times n$ matrices over $\mathbb {C}$ with $S^{2} = I_{n}$ , $(TS)^{2} = I_{n}$ . Thus $S$ and $T$ are invertible. Show that $STS^{-1} = T^{-1}$ , $ST^{-1}S = T$ .

(ii)

Let $A$ be an invertible $n \times n$ matrix over $\mathbb {C}$ and $B$ be an $n \times n$ matrix over $\mathbb {C}$ . We define the $n \times n$ matrix $D := A^{-1}BA$ . Show that $D^{n} = A^{-1}B^{n}A$ applying $AA^{-1} = I_{n}$ .

Problem 1.2.20

Find all invertible $2 \times 2$ matrices $S$ such that

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

Origin: Stated according to the original source

Problem 1.2.21

(i)(ii)

Origin: Stated according to the original source

(i)

Consider the $3 \times 2$ matrix

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

Can one find $2 \times 3$ matrices $B$ such that $AB = I_{2}$ ?

(ii)

Consider the $3 \times 2$ matrix

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

Can one find $2 \times 3$ matrices $Y$ such that $XY = I_{2}$ ?

Problem 1.2.22

Let $\overrightarrow {x} \in \mathbb {R}^{3}$ and $\times$ be the vector product. Find all solutions of

\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \times \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \times \begin{pmatrix} y_{1} \\ y_{2} \\ y_{3} \end{pmatrix} = \begin{pmatrix} 1/\sqrt{2} \\ 0 \\ 1/\sqrt{2} \end{pmatrix}.

Origin: Stated according to the original source

Problem 1.2.23

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

Origin: Stated according to the original source

(i)

Let $A$ be an $3 \times 3$ matrix over $\mathbb {R}$ and $\overrightarrow {u}, \overrightarrow {v} \in \mathbb {R}^{3}$ . Find the conditions on $A$ , $\overrightarrow {u}$ , $\overrightarrow {v}$ such that

A(\overrightarrow {u} \times \overrightarrow {v}) = (A\overrightarrow {u}) \times (A\overrightarrow {v}).

(ii)

Consider the three vectors $\overrightarrow {v}_{1}, \overrightarrow {v}_{2}, \overrightarrow {v}_{3}$ in $\mathbb {R}^{3}$ . Show that $\overrightarrow {v}_{1} \cdot (\overrightarrow {v}_{2} \times \overrightarrow {v}_{3}) = 0$ if $\overrightarrow {v}_{1}, \overrightarrow {v}_{2}, \overrightarrow {v}_{3}$ are linearly dependent.

(iii)

Let $\overrightarrow {u}, \overrightarrow {v} \in \mathbb {R}^{3}$ . Show that $(\overrightarrow {u} \times \overrightarrow {v}) \cdot (\overrightarrow {u} \times \overrightarrow {v}) \equiv (\overrightarrow {u} \cdot \overrightarrow {u})(\overrightarrow {v} \cdot \overrightarrow {v})-(\overrightarrow {u} \cdot \overrightarrow {v})^{2}$ .

(iv)

Let $\overrightarrow {a}$ , $\overrightarrow {b}$ , $\overrightarrow {c}$ , $\overrightarrow {d}$ be vectors in $\mathbb {R}^{3}$ . Show that (Lagrange identity)

(\overrightarrow {a} \times \overrightarrow {b}) \cdot (\overrightarrow {c} \times \overrightarrow {d}) \equiv \det \begin{pmatrix} \overrightarrow {a} \cdot \overrightarrow {c} & \overrightarrow {b} \cdot \overrightarrow {c} \\ \overrightarrow {a} \cdot \overrightarrow {d} & \overrightarrow {b} \cdot \overrightarrow {d} \end{pmatrix}.

Problem 1.2.24

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

Origin: Stated according to the original source

(i)

Consider four nonzero vectors $\overrightarrow {v}_{1}, \overrightarrow {v}_{2}, \overrightarrow {v}_{3}, \overrightarrow {v}_{4}$ in $\mathbb {R}^{3}$ . Let

\overrightarrow {w} := (\overrightarrow {v}_{1} \times \overrightarrow {v}_{2}) \times (\overrightarrow {v}_{3} \times \overrightarrow {v}_{4}) \neq \overrightarrow {0}.

Find $\overrightarrow {w} \times (\overrightarrow {v}_{1} \times \overrightarrow {v}_{2})$ and $\overrightarrow {w} \times (\overrightarrow {v}_{3} \times \overrightarrow {v}_{4})$ . Discuss. Note that $\overrightarrow {v}_{1}$ and $\overrightarrow {v}_{2}$ span a plane in $\mathbb {R}^{3}$ and $\overrightarrow {v}_{3}$ and $\overrightarrow {v}_{4}$ span a plane in $\mathbb {R}^{3}$ .

(ii)

Let $\overrightarrow {u}, \overrightarrow {v}, \overrightarrow {w}$ be vectors in $\mathbb {R}^{3}$ . Show that (Jacobi identity)

\overrightarrow {u} \times (\overrightarrow {v} \times \overrightarrow {w})+\overrightarrow {w} \times (\overrightarrow {u} \times \overrightarrow {v})+\overrightarrow {v} \times (\overrightarrow {w} \times \overrightarrow {u}) = \overrightarrow {0}.

(iii)

Consider the normalized column vectors

\overrightarrow {v}_{1} = \begin{pmatrix} 0 & 1 & 0 \end{pmatrix}^{T}, \quad \overrightarrow {v}_{2} = \begin{pmatrix} 1 & 0 & 1 \end{pmatrix}^{T}.

Do the three vectors $\overrightarrow {v}_{1}, \overrightarrow {v}_{2}, \overrightarrow {v}_{1} \times \overrightarrow {v}_{2}$ form an orthonormal basis in $\mathbb {R}^{3}$ ?

(iv)

Let $\overrightarrow {v}_{1}, \overrightarrow {v}_{2}, \overrightarrow {v}_{3}, \overrightarrow {v}_{4}$ be column vectors in $\mathbb {R}^{3}$ . Show that

(\overrightarrow {v}_{1} \times \overrightarrow {v}_{2})^{T}(\overrightarrow {v}_{3} \times \overrightarrow {v}_{4}) \equiv (\overrightarrow {v}_{1}^{T}\overrightarrow {v}_{3})(\overrightarrow {v}_{2}^{T}\overrightarrow {v}_{4})-(\overrightarrow {v}_{2}^{T}\overrightarrow {v}_{3})(\overrightarrow {v}_{1}^{T}\overrightarrow {v}_{4}).

A special case with $\overrightarrow {v}_{3} = \overrightarrow {v}_{1}$ , $\overrightarrow {v}_{4} = \overrightarrow {v}_{2}$ is

(\overrightarrow {v}_{1} \times \overrightarrow {v}_{2})^{T}(\overrightarrow {v}_{1} \times \overrightarrow {v}_{2}) \equiv (\overrightarrow {v}_{1}^{T}\overrightarrow {v}_{1})(\overrightarrow {v}_{2}^{T}\overrightarrow {v}_{2})-(\overrightarrow {v}_{2}^{T}\overrightarrow {v}_{1})(\overrightarrow {v}_{1}^{T}\overrightarrow {v}_{2}).

Problem 1.2.25

Consider the coordinates

\overrightarrow {p}_{1} = (x_{1},y_{1},z_{1})^{T}, \quad \overrightarrow {p}_{2} = (x_{2},y_{2},z_{2})^{T}, \quad \overrightarrow {p}_{3} = (x_{3},y_{3},z_{3})^{T}

with $\overrightarrow {p}_{1} \neq \overrightarrow {p}_{2}$ , $\overrightarrow {p}_{2} \neq \overrightarrow {p}_{3}$ , $\overrightarrow {p}_{3} \neq \overrightarrow {p}_{1}$ . We form the vectors

\overrightarrow {v}_{21} = \begin{pmatrix} x_{2}-x_{1} \\ y_{2}-y_{1} \\ z_{2}-z_{1} \end{pmatrix}, \quad \overrightarrow {v}_{31} = \begin{pmatrix} x_{3}-x_{1} \\ y_{3}-y_{1} \\ z_{3}-z_{1} \end{pmatrix}.

What does $\frac{1}{2}\left|\overrightarrow {v}_{21} \times \overrightarrow {v}_{31}\right|$ calculate? Apply it to $\overrightarrow {p}_{1} = (0,0,0)^{T}$ , $\overrightarrow {p}_{2} = (1,0,1)^{T}$ , $\overrightarrow {p}_{3} = (1,1,1)^{T}$ .

Origin: Stated according to the original source

Problem 1.2.26

Consider the $4 \times 4$ Haar matrix

K = \frac{1}{2} \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ \sqrt{2} & -\sqrt{2} & 0 & 0 \\ 0 & 0 & \sqrt{2} & -\sqrt{2} \end{pmatrix}.

Find all $4 \times 4$ hermitian matrices $H$ such that $KHK^{T} = H$ .

Origin: Stated according to the original source

Problem 1.2.27

Consider the real symmetric $3 \times 3$ matrix

A = \begin{pmatrix} -1/2 & -\sqrt{3}/6 & \sqrt{6}/3 \\ -\sqrt{3}/6 & -5/6 & -\sqrt{2}/3 \\ \sqrt{6}/3 & -\sqrt{2}/3 & 1/3 \end{pmatrix}.

Show that $A^{T} = A^{-1}$ by showing that the column of the matrix are normalized and pairwise orthonormal.

Origin: Stated according to the original source

Problem 1.2.28

Let $X \in \mathbb {R}^{n \times n}$ . Show that $X$ can be written as

X = A+S+cI_{n}

where $A$ is antisymmetric ( $A^{T} = -A$ ), $S$ is symmetric ( $S^{T} = S$ ) with $\operatorname {tr}(S) = 0$ and $c \in \mathbb {R}$ .

Origin: Stated according to the original source

Problem 1.2.29

Let $c_{j} > 0$ for $j = 1,\ldots ,n$ . Show that the $n \times n$ matrices

\left(\frac{\sqrt{c_{j}c_{k}}}{c_{j}+c_{k}}\right), \quad \left(\frac{1/c_{j}+1/c_{k}}{\sqrt{c_{j}c_{k}}}\right)

( $k = 1,\ldots ,n$ ) are positive definite.

Origin: Stated according to the original source

Problem 1.2.30

Let $a,b \in \mathbb {R}$ and $c^{2} := a^{2}+b^{2}$ with $c^{2} > 0$ . Consider the $2 \times 2$ matrix

M(a,b) = \frac{1}{c} \begin{pmatrix} a & b \\ b & -a \end{pmatrix}.

Show that $M(a,b)M(a,b) = I_{2}$ . Thus $M(a,b)$ can be considered as a square root of $I_{2}$ .

Origin: Stated according to the original source

Problem 1.2.31

Show that a nonzero $3 \times 3$ matrix $A$ over $\mathbb {R}$ which satisfies

A^{2}A^{T}+A^{T}A^{2} = 2A, \quad AA^{T}A = 2A, \quad A^{3} = 0_{3}

is given by

A = \sqrt{2} \begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \Rightarrow A^{T} = \sqrt{2} \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}.

Origin: Stated according to the original source

Problem 1.2.32

Let $\overrightarrow {x}$ be a normalized column vector in $\mathbb {R}^{n}$ , i.e. $\overrightarrow {x}^{T}\overrightarrow {x} = 1$ . A matrix $T$ is called a Householder matrix if $T := I_{n}-2\overrightarrow {x}\overrightarrow {x}^{T}$ . Show that

T^{2} = (I_{n}-2\overrightarrow {x}\overrightarrow {x}^{T})(I_{n}-2\overrightarrow {x}\overrightarrow {x}^{T}) = I_{n}-4\overrightarrow {x}\overrightarrow {x}^{T}+4\overrightarrow {x}\overrightarrow {x}^{T} = I_{n}.

Origin: Stated according to the original source

Problem 1.2.33

Let $A$ be an $n \times n$ matrix over $\mathbb {R}$ . Show that there exists nonnull vectors $\overrightarrow {x}_{1}$ , $\overrightarrow {x}_{2}$ in $\mathbb {R}^{n}$ such that

\frac{\overrightarrow {x}_{1}^{T}A\overrightarrow {x}_{1}}{\overrightarrow {x}_{1}^{T}\overrightarrow {x}_{1}} \leq \frac{\overrightarrow {x}^{T}A\overrightarrow {x}}{\overrightarrow {x}^{T}\overrightarrow {x}} \leq \frac{\overrightarrow {x}_{2}^{T}A\overrightarrow {x}_{2}}{\overrightarrow {x}_{2}^{T}\overrightarrow {x}_{2}}

for every nonnull vector $\overrightarrow {x}$ in $\mathbb {R}^{n}$ .

Origin: Stated according to the original source

Problem 1.2.34

A generalized Kronecker delta can be defined as follows

\delta _{I,J} := \begin{cases} 1 & \text{if } J = (j_{1},\ldots ,j_{r}) \text{ is an even permutation of } I = (i_{1},\ldots ,i_{r}) \\ -1 & \text{if } J \text{ is an odd permutation of } I \\ 0 & \text{if } J \text{ is not a permutation of } I \end{cases}

Show that $\delta_{126,621} = -1$ , $\delta_{126,651} = 0$ , $\delta_{125,512} = 1$ . Give a computer algebra implementation.

Origin: Stated according to the original source

Problem 1.2.35

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

Let $j_{0},j_{1},k_{0},k_{1} \in \left\{ 0,1\right\}$ . Consider the tensor

T^{j_{0},j_{1}}_{k_{0},k_{1}}.

Give a $1-1$ map that maps $T^{j_{0},j_{1}}_{k_{0},k_{1}}$ to a $2^{2} \times 2^{2}$ matrix $S = (s_{\ell_{0},\ell_{1}})$ with $\ell_{0},\ell_{1} = 0,1,2,3$ , i.e.

\begin{pmatrix} t^{00}_{00} & t^{00}_{01} & t^{00}_{10} & t^{00}_{11} \\ t^{01}_{00} & t^{01}_{01} & t^{01}_{10} & t^{01}_{11} \\ t^{10}_{00} & t^{10}_{01} & t^{10}_{10} & t^{10}_{11} \\ t^{11}_{00} & t^{11}_{01} & t^{11}_{10} & t^{11}_{11} \end{pmatrix} \mapsto \begin{pmatrix} s_{00} & s_{01} & s_{02} & s_{03} \\ s_{10} & s_{11} & s_{12} & s_{13} \\ s_{20} & s_{21} & s_{22} & s_{23} \\ s_{30} & s_{31} & s_{32} & s_{33} \end{pmatrix}.

(ii)

Let $j_{0},j_{1},j_{2},k_{0},k_{1},k_{2} \in \left\{ 0,1\right\}$ . Consider the tensor

T^{j_{0},j_{1},j_{2}}_{k_{0},k_{1},k_{2}}.

Give a $1-1$ map that maps $T^{j_{0},j_{1},j_{2}}_{k_{0},k_{1},k_{2}}$ to a $2^{3} \times 2^{3}$ matrix $S = (s_{\ell_{0},\ell_{1}})$ with $\ell_{0},\ell_{1} = 0,1,\ldots ,2^{3}-1$ .

(iii)

Let $n \geq 2$ and $j_{0},j_{1},\ldots ,j_{n},k_{0},k_{1},\ldots ,k_{n} \in \left\{ 0,1\right\}$ . Consider the tensor

T^{j_{0},j_{1},\ldots ,j_{n-1}}_{k_{0},k_{1},\ldots ,k_{n-1}}.

Give a $1-1$ map that maps the tensor to a $2^{n} \times 2^{n}$ matrix $S = (s_{\ell_{0},\ell_{1}})$ with $\ell_{0},\ell_{1} = 0,1,\ldots ,2^{n}-1$ . Give a SymbolicC++ implementation. The user provides $n$ .