Math Problems Vault

Problem 1.2.1

(i)(ii)(iii)

?

(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)

?

(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)

?

(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.

?

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 $m \in \mathbb {N}$ .

?

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 $A$ .

?

Problem 1.2.7

(i)(ii)

?

(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.

?

Problem 1.2.9

(i)(ii)

?

(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}.

?

Problem 1.2.11

(i)(ii)

?

(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.

?

Problem 1.2.13

(i)(ii)

?

(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 $\operatorname {char}(\mathbb {F}) = 2$ .

?

Problem 1.2.15

?

(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}).

?

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?

?

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.

?

(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)

?

(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}.

?

Problem 1.2.21

(i)(ii)

?

(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}.

?

Problem 1.2.23

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

?

(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)

?

(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 $\overrightarrow {p}_{1} = (0,0,0)^{T}$ , $\overrightarrow {p}_{2} = (1,0,1)^{T}$ , $\overrightarrow {p}_{3} = (1,1,1)^{T}$ .

?

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 $KHK^{T} = H$ .

?

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.

?

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 $c \in \mathbb {R}$ .

?

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.

?

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 $I_{2}$ .

?

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}.

?

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}.

?

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 $\mathbb {R}^{n}$ .

?

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.

?

Problem 1.2.35

(i)(ii)(iii)

?

(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 $n$ .