All textbooks/Problems and Solutions in Introductory and Advanced Matrix Calculus

Chapter 4

Traces, Determinants and Hyperdeterminants

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4.1 · Chapter Problems

4.2 · Supplementary Problems

Section 4.1Chapter Problems[82/100%]

Problem 4.1.1

Consider the Hadamard matrix

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

Find $\operatorname {tr}(U)$ , $\operatorname {tr}(U^{2})$ , $\operatorname {det}\left(U\right)$ and $\operatorname {det}\left(U^{2}\right)$ .

Origin: Stated according to the original source

Problem 4.1.2

Consider the $2 \times 2$ nonnormal matrix

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

Can we find an invertible $2 \times 2$ matrix $Q$ such that $Q^{-1}AQ$ is a diagonal matrix?

Origin: Stated according to the original source

Problem 4.1.3

Let $A$ be a $2 \times 2$ matrix over $\mathbb {R}$ . Assume that $\operatorname {tr}(A) = 0$ and $\operatorname {tr}(A^{2}) = 0$ . Can we conclude that $A$ is the $2 \times 2$ zero matrix?

Origin: Stated according to the original source

Problem 4.1.4

For an integer $n \geq 3$ , let $\theta := 2\pi /n$ . Find the determinant of the $n \times n$ matrix $A+I_{n}$ , where the matrix $A = (a_{jk})$ has the entries $a_{jk} = \cos (j\theta +k\theta )$ for all $j, k = 1, \ldots , n$ .

Origin: Stated according to the original source

Problem 4.1.5

Let $\alpha$ , $\beta$ , $\gamma$ , $\delta$ be real numbers. Is the $2 \times 2$ matrix

U = e^{i\alpha } \begin{pmatrix} e^{-i\beta /2} & 0 \\ 0 & e^{i\beta /2} \end{pmatrix} \begin{pmatrix} \cos (\gamma /2) & -\sin (\gamma /2) \\ \sin (\gamma /2) & \cos (\gamma /2) \end{pmatrix} \begin{pmatrix} e^{-i\delta /2} & 0 \\ 0 & e^{i\delta /2} \end{pmatrix}

unitary? What the determinant of $U$ ?

Origin: Stated according to the original source

Problem 4.1.6

Let $A$ and $B$ be two $n \times n$ matrices over $\mathbb {C}$ . If there exists a nonsingular $n \times n$ matrix $X$ such that $A = XBX^{-1}$ , then $A$ and $B$ are said to be similar matrices. Show that the spectra (eigenvalues) of two similar matrices are equal.

Origin: Stated according to the original source

Problem 4.1.7

Let $A$ and $B$ be $n \times n$ matrices over $\mathbb {C}$ . Show that the matrices $AB$ and $BA$ have the same set of eigenvalues.

Origin: Stated according to the original source

Problem 4.1.8

Let $A$ , $B$ be $n \times n$ matrices. Assume that $[A,B] = A$ . What can be said about the trace of $A$ ?

Origin: Stated according to the original source

Problem 4.1.9

An $n \times n$ matrix $A$ is called reducible if there is a permutation matrix $P$ such that

P^{T}AP = \begin{pmatrix} B & C \\ 0 & D \end{pmatrix}

where $B$ and $D$ are square matrices of order at least $1$ . An $n \times n$ matrix $A$ is called irreducible if it is not reducible. Show that the $n \times n$ primary permutation matrix

A := \begin{pmatrix} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & 1 \\ 1 & 0 & 0 & \ldots & 0 \end{pmatrix}

is irreducible.

Origin: Stated according to the original source

Problem 4.1.10

(i)(ii)(iii)

We define a linear bijection, $h$ , between $\mathbb {R}^{4}$ and $\overrightarrow {H}(2)$ , the set of complex $2 \times 2$ hermitian matrices, by

(t,x,y,z) \rightarrow \begin{pmatrix} t+x & y-iz \\ y+iz & t-x \end{pmatrix}.

We denote the matrix on the right-hand side by $H$ .

Origin: Stated according to the original source

(i)

Show that the matrix can be written as a linear combination of the Pauli spin matrices $\sigma_{1}$ , $\sigma_{2}$ , $\sigma_{3}$ and the identity matrix $I_{2}$ .

(ii)

Find the inverse map.

(iii)

Calculate the determinant of $2 \times 2$ hermitian matrix $H$ . Discuss.

Problem 4.1.11

Let $A$ be an $n \times n$ invertible matrix over $\mathbb {C}$ . Assume that $A$ can be written as $A = B+iB$ , where $B$ has only real coefficients. Show that $B^{-1}$ exists and

A^{-1} = \frac{1}{2}(B^{-1}-iB^{-1}).

Origin: Stated according to the original source

Problem 4.1.12

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

Let $A$ be an invertible matrix. Assume that $A = A^{-1}$ . What are the possible values for $\operatorname {det}\left(A\right)$ ?

(ii)

Let $B$ be a skew-symmetric matrix over $\mathbb {R}$ , i.e. $A^{T} = -B$ and of order $2n-1$ . Show that $\operatorname {det}\left(B\right) = 0$ .

(iii)

Show that if a square matrix $C$ is hermitian, i.e. $C^{*} = C$ then $\operatorname {det}\left(C\right)$ is a real number.

Problem 4.1.13

(i)(ii)

Origin: Stated according to the original source

(i)

Let $A$ , $B$ are $2 \times 2$ matrices over $\mathbb {R}$ . Let $H := A+iB$ . Express $\operatorname {det}\left(H\right)$ as a sum of determinants.

(ii)

Let $A$ , $B$ are $2 \times 2$ matrices over $\mathbb {R}$ . Let $H := A+iB$ . Assume that $H$ is hermitian. Show that $\operatorname {det}\left(H\right) = \operatorname {det}\left(A\right)-\operatorname {det}\left(B\right)$ .

Problem 4.1.14

(i)(ii)

Origin: Stated according to the original source

(i)

Let $A$ , $B$ , and $C$ be $n \times n$ matrices. Calculate

\operatorname {det}\left(\begin{array}{ll} A & 0_{n} \\ C & B \end{array}\right).

(ii)

Let $A$ , $B$ , $C$ , $D$ be $n \times n$ matrices. Assume that $DC = CD$ , i.e. $C$ and $D$ commute and $\operatorname {det}\left(D\right) \neq 0$ . Consider the $(2n) \times (2n)$ matrix

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

Show that

\operatorname {det}\left(M\right) = \operatorname {det}\left(AD-BC\right). \tag {1}

We know that

\operatorname {det}\left(\begin{array}{ll} U & 0_{n} \\ X & Y \end{array}\right) = \operatorname {det}\left(U\right)\operatorname {det}\left(Y\right), \qquad \operatorname {det}\left(\begin{array}{ll} U & V \\ 0_{n} & Y \end{array}\right) = \operatorname {det}\left(U\right)\operatorname {det}\left(Y\right) \tag {2}

where $U$ , $V$ , $X$ , $Y$ are $n \times n$ matrices.

Problem 4.1.15

Let $A$ , $B$ be $n \times n$ matrices. We have the identity

\operatorname {det}\left(\begin{array}{ll} A & B \\ B & A \end{array}\right) \equiv \operatorname {det}\left(A+B\right)\operatorname {det}\left(A-B\right).

Use this identity to calculate the determinant of the left-hand side using the right-hand side, where

A = \begin{pmatrix} 2 & 3 \\ 1 & 7 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 2 \\ 4 & 6 \end{pmatrix}.

Origin: Stated according to the original source

Problem 4.1.16

(i)(ii)

Origin: Stated according to the original source

(i)

Let $A$ , $B$ be $n \times n$ matrices. Show that

\operatorname {tr}((A+B)(A-B)) \equiv \operatorname {tr}(A^{2})-\operatorname {tr}(B^{2}).

(ii)

Let $A$ , $B$ , $C$ be $n \times n$ matrices. Show that $\operatorname {tr}([A,B]C) = \operatorname {tr}(A[B,C])$ .

Problem 4.1.17

Let $A$ , $B$ be $n \times n$ positive definite matrices. Show that $\operatorname {tr}(AB) > 0$ .

Origin: Stated according to the original source

Problem 4.1.18

Let

A = \begin{pmatrix} \dfrac {1}{x_{1}+y_{1}} & \dfrac {1}{x_{1}+y_{2}} \\ ]2ex] \dfrac {1}{x_{2}+y_{1}} & \dfrac {1}{x_{2}+y_{2}} \end{pmatrix} \[ where we assume that $x_{i}+y_{j} \neq 0$ for $i,j = 1,2$. Show that \] \operatorname {det}\left(A\right) = \frac{(x_{1}-x_{2})(y_{1}-y_{2})}{(x_{1}+y_{1})(x_{1}+y_{2})(x_{2}+y_{1})(x_{2}+y_{2})}. \[ \]

Origin: Stated according to the original source

Problem 4.1.19

(i)(ii)

For a $3 \times 3$ matrix we can use the rule of Sarrus to calculate the determinant. Let

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

Write the first two columns again to the right of the matrix to obtain

\begin{pmatrix} {[ccc|cc]} a_{11} & a_{12} & a_{13} & a_{11} & a_{12} \\ a_{21} & a_{22} & a_{23} & a_{21} & a_{22} \\ a_{31} & a_{32} & a_{33} & a_{31} & a_{32} \end{pmatrix}.

Now look at the diagonals. The product of the diagonals sloping down to the right have a plus sign, the ones up to the left have a negative sign. This leads to the determinant

\operatorname {det}\left(A\right) = a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{31}a_{22}a_{13} - a_{32}a_{23}a_{11} - a_{33}a_{21}a_{12}.

Origin: Stated according to the original source

(i)

Use this rule to calculate the determinant of the rotational matrix

R(\theta ) = \begin{pmatrix} \cos (\theta ) & 0 & -\sin (\theta ) \\ 0 & 1 & 0 \\ \sin (\theta ) & 0 & \cos (\theta ) \end{pmatrix}.

(ii)

Use the rule to find the determinant of the matrix

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

Problem 4.1.20

Let $A$ , $S$ be $n \times n$ matrices. Assume that $S$ is invertible and assume that $S^{-1}AS = \rho S$ , where $\rho \neq 0$ . Show that $A$ is invertible.

Origin: Stated according to the original source

Problem 4.1.21

Let $A$ be an invertible $n \times n$ matrix. Let $c=2$ . Can we find an invertible matrix $S$ such that $SAS^{-1} = cA$ ?

Origin: Stated according to the original source

Problem 4.1.22

The determinant of an $n \times n$ circulant matrix is given by

\operatorname {det}\left(\begin{array}{lllll} a_{1} & a_{2} & a_{3} & \ldots & a_{n} \\ a_{n} & a_{1} & a_{2} & \ldots & a_{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{3} & a_{4} & a_{5} & & a_{2} \\ a_{2} & a_{3} & a_{4} & & a_{1} \end{array}\right) = (-1)^{n-1} \prod _{j=0}^{n-1} \left(\sum _{k=1}^{n} \zeta ^{jk}a_{k}\right) \tag {1}

where $\zeta := \exp (2\pi i/n)$ . Find the determinant of the circulant $n \times n$ matrix

C = \begin{pmatrix} 1 & 4 & 9 & & n^{2} \\ n^{2} & 1 & 4 & & (n-1)^{2} \\ \vdots & \vdots & \vdots & & \vdots \\ 9 & 16 & 25 & \ldots & 4 \\ 4 & 9 & 16 & \ldots & 1 \end{pmatrix}

using equation (1).

Origin: Stated according to the original source

Problem 4.1.23

Let $A$ be a nonzero $2 \times 2$ matrix over $\mathbb {R}$ . Let $B_{1}$ , $B_{2}$ , $B_{3}$ , $B_{4}$ be $2 \times 2$ matrices over $\mathbb {R}$ and assume that

\operatorname {det}\left(A+B_{j}\right) = \operatorname {det}\left(A\right)+\operatorname {det}\left(B_{j}\right) \qquad \text{for } j=1,2,3,4.

Show that there exist real numbers $c_{1}$ , $c_{2}$ , $c_{3}$ , $c_{4}$ , not all zero, such that

c_{1}B_{1}+c_{2}B_{2}+c_{3}B_{3}+c_{4}B_{4} = 0_{2}. \tag {1}

Origin: Stated according to the original source

Problem 4.1.24

(i)(ii)

An $n \times n$ matrix $Q$ is orthogonal if $Q$ is real and $Q^{T}Q = Q^{T}Q = I_{n}$ i.e. $Q^{-1} = Q^{T}$ .

Origin: Stated according to the original source

(i)

Find the determinant of an orthogonal matrix.

(ii)

Let $\overrightarrow {u}$ , $\overrightarrow {v}$ be two vectors in $\mathbb {R}^{3}$ and $\overrightarrow {u} \times \overrightarrow {v}$ denotes the vector product of $\overrightarrow {u}$ and $\overrightarrow {v}$

\overrightarrow {u} \times \overrightarrow {v} := \begin{pmatrix} u_{2}v_{3}-u_{3}v_{2} \\ u_{3}v_{1}-u_{1}v_{3} \\ u_{1}v_{2}-u_{2}v_{1} \end{pmatrix}.

Let $Q$ be a $3 \times 3$ orthogonal matrix. Calculate $(Q\overrightarrow {u}) \times (Q\overrightarrow {v})$ .

Problem 4.1.25

(i)(ii)

Origin: Stated according to the original source

(i)

Calculate the determinant of the $n \times n$ matrix

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

(ii)

Find the determinant of the matrix

C = \begin{pmatrix} 0 & 1 & 1 & & 1 & 1 \\ 1 & 0 & 1 & & 1 & 1 \\ 1 & 1 & 0 & & 1 & 1 \\ \vdots & \vdots & \vdots & & \vdots & \vdots \\ 1 & 1 & 1 & \ldots & 0 & 1 \\ 1 & 1 & 1 & \ldots & 1 & 0 \end{pmatrix}.

Problem 4.1.26

Let $A$ be a $2 \times 2$ matrix over $\mathbb {R}$ with $\operatorname {det}\left(A\right) \neq 0$ . Is $(A^{T})^{-1} = (A^{-1})^{T}$ ?

Origin: Stated according to the original source

Problem 4.1.27

Find all $2 \times 2$ matrices $A$ over $\mathbb {C}$ that satisfy the three conditions

\operatorname {tr}(A) = 0, \qquad A = A^{*}, \qquad A^{2} = I_{2}.

Origin: Stated according to the original source

Problem 4.1.28

Let $A$ be an $n \times n$ matrix with $A^{2} = I_{n}$ . Let $B$ be a matrix with $AB = -BA$ , i.e. $[A,B]_{+} = 0_{n}$ . Show that $\operatorname {tr}(B) = 0$ . Find $\operatorname {tr}(A \otimes B)$ .

Origin: Stated according to the original source

Problem 4.1.29

Consider the two $2 \times 2$ matrices

A = \begin{pmatrix} a_{11} & 1 \\ a_{21} & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} a_{11} & 0 \\ a_{21} & 1 \end{pmatrix}.

The first column of the matrices $A$ and $B$ agree, but the second column of the two matrices differ. Is $\operatorname {det}\left(A+B\right) = 2(\operatorname {det}\left(A\right)+\operatorname {det}\left(B\right))$ ?

Origin: Stated according to the original source

Problem 4.1.30

Let $A$ , $B$ be $2 \times 2$ matrices. Assume that $\operatorname {det}\left(A\right) = 0$ and $\operatorname {det}\left(B\right) = 0$ . Can we conclude that $\operatorname {det}\left(A+B\right) = 0$ ?

Origin: Stated according to the original source

Problem 4.1.31

Consider the symmetric $3 \times 3$ matrix

A = \begin{pmatrix} n_{1} & 1 & 0 \\ 1 & n_{2} & 1 \\ 0 & 1 & n_{3} \end{pmatrix}.

Find all positive integers $n_{1}$ , $n_{2}$ , $n_{3}$ such that $\operatorname {det}\left(A\right) = 1$ and $A$ is positive definite.

Origin: Stated according to the original source

Problem 4.1.32

(i)(ii)

The oriented volume of an $n$ -simplex in $n$ -dimensional Euclidean space with vertices $\overrightarrow {v}_{0}, \overrightarrow {v}_{1}, \ldots , \overrightarrow {v}_{n}$ is given by

\frac{1}{n!} \operatorname {det}\left(S\right)

where $S$ is the $n \times n$ matrix $S := \begin{pmatrix} \overrightarrow {v}_{1}-\overrightarrow {v}_{0} & \overrightarrow {v}_{2}-\overrightarrow {v}_{0} & \ldots & \overrightarrow {v}_{n-1}-\overrightarrow {v}_{0} & \overrightarrow {v}_{n}-\overrightarrow {v}_{0} \end{pmatrix}$ . Thus each column of the $n \times n$ matrix is the difference between the vectors representing two vertices.

Origin: Stated according to the original source

(i)

Let

\overrightarrow {v}_{0} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}, \qquad \overrightarrow {v}_{1} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \qquad \overrightarrow {v}_{2} = \begin{pmatrix} 1/2 \\ 1 \end{pmatrix}.

Find the oriented volume.

(ii)

Let

\overrightarrow {v}_{0} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}, \qquad \overrightarrow {v}_{1} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \qquad \overrightarrow {v}_{2} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \qquad \overrightarrow {v}_{3} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}.

Find the oriented volume.

Problem 4.1.33

(i)(ii)(iii)

The area $A$ of a triangle given by the coordinates of its vertices

(x_{0}, y_{0}), \qquad (x_{1}, y_{1}), \qquad (x_{2}, y_{2})

A = \frac{1}{2} \operatorname {det}\left(\begin{array}{lll} x_{0} & y_{0} & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{array}\right).

Origin: Stated according to the original source

(i)

Let $(x_{0},y_{0}) = (0,0)$ , $(x_{1},y_{1}) = (1,0)$ , $(x_{2},y_{2}) = (0,1)$ . Find $A$ .

(ii)

A tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A tetrahedron can be defined by the coordinates of the vertices

(x_{0},y_{0},z_{0}), \quad (x_{1},y_{1},z_{1}), \quad (x_{2},y_{2},z_{2}), \quad (x_{3},y_{3},z_{3}).

The volume $V$ of the tetrahedron is given by (actually $\left|V\right|$ )

V = \frac{1}{6} \operatorname {det}\left(\begin{array}{llll} x_{0} & y_{0} & z_{0} & 1 \\ x_{1} & y_{1} & z_{1} & 1 \\ x_{2} & y_{2} & z_{2} & 1 \\ x_{3} & y_{3} & z_{3} & 1 \end{array}\right).

Let

(x_{0},y_{0},z_{0}) = (0,0,0), \quad (x_{1},y_{1},z_{1}) = (0,0,1),

(x_{2},y_{2},z_{2}) = (0,1,0), \quad (x_{3},y_{3},z_{3}) = (1,0,0).

Find the volume $V$ .

(iii)

Let

(x_{0},y_{0},z_{0}) = (+1,+1,+1), \quad (x_{1},y_{1},z_{1}) = (-1,-1,+1),

(x_{2},y_{2},z_{2}) = (-1,+1,-1), \quad (x_{3},y_{3},z_{3}) = (+1,-1,-1).

Find the volume $V$ .

Problem 4.1.34

(i)(ii)

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Assume that $\operatorname {tr}(AB) = 0$ .

Origin: Stated according to the original source

(i)

Can we conclude that $\operatorname {tr}(AB^{*}) = 0$ ?

(ii)

Consider the case that $B$ is skew-hermitian.

Problem 4.1.35

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Is $\operatorname {tr}(AB^{*}) = \operatorname {tr}(A^{*}B)$ ?

Origin: Stated according to the original source

Problem 4.1.36

Let

A = (\overrightarrow {a}_{1}, \overrightarrow {a}_{2}, \ldots , \overrightarrow {a}_{n-1}, \overrightarrow {u}), \qquad B = (\overrightarrow {a}_{1}, \overrightarrow {a}_{2}, \ldots , \overrightarrow {a}_{n-1}, \overrightarrow {v})

be $n \times n$ matrices, where the first $n-1$ columns $\overrightarrow {a}_{1}, \ldots , \overrightarrow {a}_{n-1}$ are the same and for the last column $\overrightarrow {u} \neq \overrightarrow {v}$ . Show that

\operatorname {det}\left(A+B\right) = 2^{n-1}(\operatorname {det}\left(A\right)+\operatorname {det}\left(B\right)).

Origin: Stated according to the original source

Problem 4.1.37

Find all $2 \times 2$ matrices $g$ over $\mathbb {C}$ such that $\operatorname {det}\left(g\right) = 1$ , $\eta g^{*} \eta = g^{-1}$ , where $\eta$ is the diagonal matrix $\eta = \operatorname {diag}(1,-1)$ .

Origin: Stated according to the original source

Problem 4.1.38

An $n \times n$ tridiagonal matrix ( $n \geq 3$ ) has nonzero elements only in the main diagonal, the first diagonal below this, and the first diagonal above the main diagonal. The determinant of an $n \times n$ tridiagonal matrix can be calculated by the recursive formula

\operatorname {det}\left(A\right) = a_{n,n}\operatorname {det}\left(A\right)_{\left\{ 1,\ldots ,n-1\right\} } - a_{n,n-1}a_{n-1,n}\operatorname {det}\left(A\right)_{\left\{ 1,\ldots ,n-2\right\} }

where $\operatorname {det}\left(A\right)_{\left\{ 1,\ldots ,k\right\} }$ denotes the $k$ -th principal minor, that is, $[A]_{\left\{ 1,\ldots ,k\right\} }$ is the submatrix by the first $k$ rows and columns of $A$ . The cost of computing the determinant of a tridiagonal matrix using this recursion is linear in $n$ , while the cost is cubic for a general matrix. Apply this recursion relation to calculate the determinant of the $4 \times 4$ matrix

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

Origin: Stated according to the original source

Problem 4.1.39

(i)(ii)

Consider the symmetric $3 \times 3$ matrix

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

Origin: Stated according to the original source

(i)

Find the maxima and minima of the function $f(\alpha ) = \operatorname {det}\left(A(\alpha )\right)$ .

(ii)

For which values of $\alpha$ is the matrix noninvertible?

Problem 4.1.40

The Pascal matrix of order $n$ is defined as

P_{n} := \left(\frac{(i+j-2)!}{(i-1)!(j-1)!}\right), \qquad i,j = 1, \ldots , n.

Thus

P_{2} = \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}, \qquad P_{3} = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix}, \qquad P_{4} = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \\ 1 & 3 & 6 & 10 \\ 1 & 4 & 10 & 20 \end{pmatrix}.

Find the determinant of $P_{2}$ , $P_{3}$ , $P_{4}$ . Find the inverse of $P_{2}$ , $P_{3}$ , $P_{4}$ .

Origin: Stated according to the original source

Problem 4.1.41

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {R}$ . Assume that $A$ is invertible. Let $t$ be a nonzero real number. Show that

\operatorname {det}\left(A+tB\right) = t^{n}\operatorname {det}\left(A\right)\operatorname {det}\left(A^{-1}B+t^{-1}I_{n}\right).

Origin: Stated according to the original source

Problem 4.1.42

Let $A$ be an $n \times n$ invertible matrix over $\mathbb {R}$ . Show that $A^{T}$ is also invertible. Is $(A^{T})^{-1} = (A^{-1})^{T}$ ?

Origin: Stated according to the original source

Problem 4.1.43

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

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

using the exterior product $\wedge$ (wedge product). This means calculate

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

Origin: Stated according to the original source

Problem 4.1.44

The $3 \times 3$ diagonal matrices over $\mathbb {R}$ with trace equal to $0$ form a vector space. Provide a basis for this vector space. Using the scalar product $\operatorname {tr}(AB^{T})$ for $n \times n$ matrices $A, B$ over $\mathbb {R}$ the elements of the basis should be orthogonal to each other.

Origin: Stated according to the original source

Problem 4.1.45

The Hilbert-Schmidt norm of an $n \times n$ matrix over $\mathbb {C}$ is defined by

\left\| A\right\| _{2} := \sqrt{\operatorname {tr}(A^{*}A)}.

Another norm is the trace norm defined by

\left\| A\right\| _{1} := \operatorname {tr}\sqrt{(A^{*}A)}.

Calculate the two norms for the $2 \times 2$ matrix

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

Origin: Stated according to the original source

Problem 4.1.46

Let $A$ be a $3 \times 3$ matrix over $\mathbb {R}$ . Consider the $3 \times 3$ permutation matrix

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

Assume that $AP = A$ . Show that $A$ cannot be invertible.

Origin: Stated according to the original source

Problem 4.1.47

Let $A = (a_{ij})$ be a $2n \times 2n$ skew-symmetric matrix. The Pfaffian is defined as

\operatorname {Pf}(A) := \frac{1}{2^{n}n!} \sum _{\sigma \in S_{2n}} \operatorname {sgn}(\sigma ) \prod _{j=1}^{n} a_{\sigma (2j-1), \sigma (2j)}

where $S_{2n}$ is the symmetric group and $\operatorname {sgn}(\sigma )$ is the signature of permutation $\sigma$ . Consider the case with $n=2$ , i.e.

A = \begin{pmatrix} 0 & a_{12} & a_{13} & a_{14} \\ -a_{12} & 0 & a_{23} & a_{24} \\ -a_{13} & -a_{23} & 0 & a_{34} \\ -a_{14} & -a_{24} & -a_{34} & 0 \end{pmatrix}.

Calculate $\operatorname {Pf}(A)$ .

Origin: Stated according to the original source

Problem 4.1.48

Let $A$ be a skew-symmetric $2n \times 2n$ matrix. For the Pfaffian we have the properties

(\operatorname {Pf}(A))^{2} = \operatorname {det}\left(A\right), \qquad \operatorname {Pf}(BAB^{T}) = \operatorname {det}\left(B\right)\operatorname {Pf}(A)

\operatorname {Pf}(\lambda A) = \lambda ^{n}\operatorname {Pf}(A), \qquad \operatorname {Pf}(A^{T}) = (-1)^{n}\operatorname {Pf}(A)

where $B$ is an arbitrary $2n \times 2n$ matrix. Let $J$ be a $2n \times 2n$ skew-symmetric matrix with $\operatorname {Pf}(J) \neq 0$ . Let $B$ be a $2n \times 2n$ matrix such that $B^{T}JB = J$ . Show that $\operatorname {det}\left(B\right) = 1$ .

Origin: Stated according to the original source

Problem 4.1.49

Find all $2 \times 2$ invertible matrices $A$ such that $A+A^{-1} = I_{2}$ .

Origin: Stated according to the original source

Problem 4.1.50

Let $V_{1}$ be a hermitian $n \times n$ matrix. Let $V_{2}$ be a positive semidefinite $n \times n$ matrix. Let $k$ be a positive integer. Show that $\operatorname {tr}((V_{2}V_{1})^{k})$ can be written as $\operatorname {tr}(V^{k})$ , where $V := V_{2}^{1/2}V_{1}V_{2}^{1/2}$ .

Origin: Stated according to the original source

Problem 4.1.51

Consider the $2 \times 2$ matrix

M = \begin{pmatrix} \cosh (r)-\sinh (r)\cos (2\theta ) & -\sinh (r)\sin (2\theta ) \\ -\sinh (r)\sin (2\theta ) & \cosh (r)+\sinh (r)\cos (2\theta ) \end{pmatrix}.

Find the determinant of $M$ . Thus show that the inverse of $M$ exists. Find the inverse of $M$ .

Origin: Stated according to the original source

Problem 4.1.52

Consider the $2 \times 2$ matrix $C$ over $\mathbb {C}$ . Calculate $\operatorname {det}\left(CC^{*}\right)$ and show that $\operatorname {det}\left(CC^{*}\right) \geq 0$ .

Origin: Stated according to the original source

Problem 4.1.53

(i)(ii)

Origin: Stated according to the original source

(i)

Let $z \in \mathbb {C}$ . Find the determinant of

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

Is the matrix

\Pi _{2} = I_{2} - \frac{1}{1+z\overline{z}}A

a projection matrix?

(ii)

Let $z_{1}, z_{2} \in \mathbb {C}$ . Find the determinant of

B = \begin{pmatrix} 1 & z_{1} & z_{2} \\ \overline{z}_{1} & z_{1}\overline{z}_{1} & z_{2}\overline{z}_{1} \\ \overline{z}_{2} & z_{1}\overline{z}_{2} & z_{2}\overline{z}_{2} \end{pmatrix}.

Is the matrix

\Pi _{3} = I_{3} - \frac{1}{1+z_{1}\overline{z}_{1}+z_{2}\overline{z}_{2}}B

a projection matrix?

Problem 4.1.54

Consider the golden mean number $\tau = (\sqrt{5}-1)/2$ and the $2 \times 2$ matrix

F = \begin{pmatrix} \tau & \sqrt{\tau } \\ \sqrt{\tau } & -\tau \end{pmatrix}.

Find $\operatorname {tr}(F)$ and $\operatorname {det}\left(F\right)$ . Since $\operatorname {det}\left(F\right) \neq 0$ we have an inverse. Find $F^{-1}$ .

Origin: Stated according to the original source

Problem 4.1.55

Let $A$ be an $n \times n$ matrix and $B$ be an invertible $n \times n$ matrix. Show that

\operatorname {det}\left(I_{n}+A\right) = \operatorname {det}\left(I_{n}+BAB^{-1}\right).

Origin: Stated according to the original source

Problem 4.1.56

Let $A$ be an $2 \times 2$ matrix. Show that

\operatorname {det}\left(I_{2}+A\right) = 1+\operatorname {tr}(A)+\operatorname {det}\left(A\right).

Origin: Stated according to the original source

Problem 4.1.57

Let $\left\{ \overrightarrow {e}_{j} : j=1,2,3\right\}$ be the three orthonormal vectors in $\mathbb {Z}^{3}$

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

We consider the face-centered cubic lattice as a sublattice of $\mathbb {Z}^{3}$ generated by the three primitive vectors $\overrightarrow {e}_{1}+\overrightarrow {e}_{2}$ , $\overrightarrow {e}_{1}+\overrightarrow {e}_{3}$ , $\overrightarrow {e}_{2}+\overrightarrow {e}_{3}$ . Form the $3 \times 3$ matrix $\begin{pmatrix} \overrightarrow {e}_{1}+\overrightarrow {e}_{2} & \overrightarrow {e}_{1}+\overrightarrow {e}_{3} & \overrightarrow {e}_{2}+\overrightarrow {e}_{3} \end{pmatrix}$ and show that this matrix has an inverse. Find the inverse.

Origin: Stated according to the original source

Problem 4.1.58

Let $n \geq 2$ . Consider the $n \times n$ symmetric tridiagonal matrix over $\mathbb {R}$

A_{n} = \begin{pmatrix} c & 1 & 0 & 0 & \cdots & & 0 \\ 1 & c & 1 & 0 & \cdots & & 0 \\ 0 & 1 & c & 1 & \cdots & & 0 \\ \vdots & & & \ddots & & & \vdots \\ 0 & 0 & 0 & \cdots & 1 & c & 1 \\ 0 & 0 & 0 & \cdots & 0 & 1 & c \end{pmatrix}

where $c \in \mathbb {R}$ . Find the determinant of $A_{n}$ .

Origin: Stated according to the original source

Problem 4.1.59

Find a nonzero $2 \times 2$ matrix $V$ over $\mathbb {C}$ such that $V^{2} = \operatorname {tr}(V)V$ . Can such a matrix be invertible?

Origin: Stated according to the original source

Problem 4.1.60

Consider the $(n+1) \times (n+1)$ matrix over $\mathbb {C}$

A = \begin{pmatrix} 1 & 0 & 0 & & 0 & z_{1} \\ 0 & 1 & 0 & & 0 & z_{2} \\ 0 & 0 & 1 & & 0 & z_{3} \\ \vdots & \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & 0 & \ldots & 1 & z_{n} \\ z_{1} & z_{2} & z_{3} & \ldots & z_{n} & 1 \end{pmatrix}.

Find the determinant. What is the condition on the $z_{j}$ 's such that $A$ is invertible?

Origin: Stated according to the original source

Problem 4.1.61

Let $z_{k} = x_{k}+iy_{k}$ , where $x_{k}, y_{k} \in \mathbb {R}$ and $k=1,\ldots ,n$ . Find the $2n \times 2n$ matrix $A$ such that

\begin{pmatrix} z_{1} \\ \vdots \\ z_{n} \\ \overline{z}_{1} \\ \vdots \\ \overline{z}_{n} \end{pmatrix} = A \begin{pmatrix} x_{1} \\ \vdots \\ x_{n} \\ y_{1} \\ \vdots \\ y_{n} \end{pmatrix}.

Find the determinant of the matrix $A$ .

Origin: Stated according to the original source

Problem 4.1.62

(i)(ii)

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . We define the product

A \diamond B := \frac{1}{2}(AB+BA) - \frac{1}{n}\operatorname {tr}(AB)I_{n}.

Origin: Stated according to the original source

(i)

Find the trace of $A \diamond B$ .

(ii)

Is the product commutative?

Problem 4.1.63

Let $A$ be an $n \times n$ invertible matrix over $\mathbb {C}$ . Let $\overrightarrow {x}, \overrightarrow {y} \in \mathbb {C}^{n}$ . Then we have the identity

\operatorname {det}\left(A+\overrightarrow {x}\overrightarrow {y}^{*}\right) \equiv \operatorname {det}\left(A\right)(1+\overrightarrow {y}^{*}A^{-1}\overrightarrow {x}).

Can we conclude that $A+\overrightarrow {x}\overrightarrow {y}^{*}$ is also invertible?

Origin: Stated according to the original source

Problem 4.1.64

Let $n \geq 1$ . Consider the $2^{n} \times 2^{n}$ matrices $A$ and $B$ . Assume that $A^{2} = I_{2^{n}}$ , $B^{2} = I_{2^{n}}$ and $[A,B]_{+} = 0_{2^{n}}$ . Show that

\operatorname {det}\left(A+B\right) = 2^{2^{n}/2}.

Origin: Stated according to the original source

Problem 4.1.65

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Assume that $B$ is invertible. Show that there exists $c \in \mathbb {C}$ such that $A+cB$ is not invertible.

Origin: Stated according to the original source

Remark.

Origin: Stated according to the original source

Start of with the identity $A+cB \equiv (AB^{-1}+cI_{n})B$ and apply the determinant.

Problem 4.1.66

The permanent of an $n \times n$ matrix $A$ is defined as

\operatorname {Per}(A) := \sum _{\sigma \in S_{n}} \prod _{j=1}^{n} a_{j,\sigma (j)}

where $S_{n}$ is the set of all permutation of $n$ elements. Find the permanent of a $2 \times 2$ matrix $A$ .

Origin: Stated according to the original source

Problem 4.1.67

Let $A$ be a $4 \times 4$ matrix we write as

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

where $A_{1}$ , $A_{2}$ , $A_{3}$ , $A_{4}$ are the (block) $2 \times 2$ matrices in the matrix $A$ . Consider the map

\begin{pmatrix} A_{1} & A_{2} \\ A_{3} & A_{4} \end{pmatrix} \mapsto \begin{pmatrix} A_{1} & A_{2}^{T} \\ A_{3}^{T} & A_{4} \end{pmatrix}

where $^{T}$ denotes the transpose. Is the trace preserved under this map? Is the determinant preserved under this map?

Origin: Stated according to the original source

Problem 4.1.68

Let $H$ be a $4 \times 4$ matrix and $U$ , $V$ be $2 \times 2$ unitary matrices. Calculate

\operatorname {det}\left((U \otimes V)H(U^{*} \otimes V^{*})\right), \qquad \operatorname {tr}((U \otimes V)H(U^{*} \otimes V^{*})).

Origin: Stated according to the original source

Problem 4.1.69

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Consider the expression

(A \otimes B - B \otimes A) \oplus (-[A,B])

where $\oplus$ denotes the direct sum. Find the trace of this expression. This expression plays a role for universal enveloping algebras.

Origin: Stated according to the original source

Problem 4.1.70

Let $A$ be an $2 \times 2$ matrix over $\mathbb {R}$ . Let $\operatorname {tr}(A) = c_{1}$ , $\operatorname {tr}(A^{2}) = c_{2}$ . Can $\operatorname {det}\left(A\right)$ be calculated from $c_{1}$ , $c_{2}$ ?

Origin: Stated according to the original source

Problem 4.1.71

Let $J$ be the $2n \times 2n$ matrix

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

We define symplectic $\overrightarrow {G}$ -reflectors to be those $2n \times 2n$ symplectic matrices that have a $(2n-1)$ -dimensional fixed-point subspace. Any symplectic $\overrightarrow {G}$ -reflector can be expressed as

G = I_{2n} + \beta \overrightarrow {u}\overrightarrow {u}^{T}J \tag {1}

for some $0 \neq \beta \in \mathbb {F}$ , $0 \neq \overrightarrow {u} \in \mathbb {F}^{2n}$ and $\overrightarrow {u}$ is considered as a column vector. The underlying field is $\mathbb {F}$ . Conversely, any $G$ given by (1) is always a symplectic $\overrightarrow {G}$ -reflector. Show that $\operatorname {det}\left(G\right) = +1$ .

Origin: Stated according to the original source

Problem 4.1.72

Let $A$ , $B$ be positive semidefinite $n \times n$ matrices. Then

\operatorname {det}\left(A+B\right) \geq \operatorname {det}\left(A\right)+\operatorname {det}\left(B\right), \qquad \operatorname {tr}(AB) \leq \operatorname {tr}(A)\operatorname {tr}(B).

Let

A = \frac{1}{3} \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}, \qquad B = \frac{1}{3} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}.

Both matrices are density matrices, with $A$ a pure state and $B$ a mixed state. Calculate the left- and right-hand side of the two inequality. Discuss.

Origin: Stated according to the original source

Problem 4.1.73

(i)(ii)

Let $A$ be an $n \times n$ positive semidefinite matrix. Let $B$ be an $n \times n$ positive definite matrix. Then we have Klein’s inequality

\operatorname {tr}(A(\ln (A)-\ln (B))) \geq \operatorname {tr}(A-B).

Origin: Stated according to the original source

(i)

Let

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

Calculate the left-hand side and the right-hand side of the inequality.

(ii)

When is the inequality an equality?

Problem 4.1.74

Consider the two polynomials

p_{1}(x) = a_{0}+a_{1}x+\cdots +a_{n}x^{n}, \qquad p_{2}(x) = b_{0}+b_{1}x+\cdots +b_{m}x^{m}

where $n = \deg (p_{1})$ and $m = \deg (p_{2})$ . Assume that $n>m$ . Let $r(x) := p_{2}(x)/p_{1}(x)$ . We expand $r(x)$ in powers of $1/x$ , i.e.

r(x) = \frac{c_{1}}{x} + \frac{c_{2}}{x^{2}} + \cdots .

From the coefficients $c_{1}, c_{2}, \ldots , c_{2n-1}$ we can form an $n \times n$ Hankel matrix

H_{n} = \begin{pmatrix} c_{1} & c_{2} & & c_{n} \\ c_{2} & c_{3} & & c_{n+1} \\ \vdots & \vdots & \ddots & \vdots \\ c_{n} & c_{n+1} & \cdots & c_{2n-1} \end{pmatrix}.

The determinant of this matrix is proportional to the resultant of the two polynomials. If the resultant vanishes, then the two polynomials have a non-trivial greatest common divisor. Apply this theorem to the polynomials

p_{1}(x) = x^{3}+6x^{2}+11x+6, \qquad p_{2}(x) = x^{2}+4x+3.

Origin: Stated according to the original source

Problem 4.1.75

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

Consider the $5 \times 5$ symmetric matrix over $\mathbb {R}$

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

The rank is the number of linearly independent row (or column) vectors of $A$ .

Origin: Stated according to the original source

(i)

Find the rank of the matrix. Explain.

(ii)

Find the determinant and trace of the matrix.

(iii)

Find all eigenvalues of the matrix.

(iv)

Find one eigenvector.

(v)

Is the matrix positive semidefinite?

Problem 4.1.76

(i)(ii)

Let $A$ be an $n \times n$ matrix over $\mathbb {R}$ . Assume that $A^{-1}$ exists. Let $\overrightarrow {u}, \overrightarrow {v} \in \mathbb {R}^{n}$ , where $\overrightarrow {u}, \overrightarrow {v}$ are considered as column vectors.

Origin: Stated according to the original source

(i)

Show that if $\overrightarrow {v}^{T}A^{-1}\overrightarrow {u} = -1$ , then $A+\overrightarrow {u}\overrightarrow {v}^{T}$ is not invertible.

(ii)

Assume that $\overrightarrow {v}^{T}A^{-1}\overrightarrow {u} \neq -1$ . Show that

(A+\overrightarrow {u}\overrightarrow {v}^{T})^{-1} = A^{-1} - \frac{A^{-1}\overrightarrow {u}\overrightarrow {v}^{T}A^{-1}}{1+\overrightarrow {v}^{T}A^{-1}\overrightarrow {u}}. \tag {1}

Problem 4.1.77

The collocation polynomial $p(x)$ for unequally-spaced arguments $x_{0}, x_{1}, \ldots , x_{n}$ can be found by the determinant method

\operatorname {det}\left(\begin{array}{llllll} p(x) & 1 & x & x^{2} & \ldots & x^{n} \\ y_{0} & 1 & x_{0} & x_{0}^{2} & \ldots & x_{0}^{n} \\ y_{1} & 1 & x_{1} & x_{1}^{2} & \ldots & x_{1}^{n} \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ y_{n} & 1 & x_{n} & x_{n}^{2} & \ldots & x_{n}^{n} \end{array}\right) = 0

where $p(x_{k}) = y_{k}$ for $k=0,1,\ldots ,n$ . Apply it to ( $n=2$ )

p(x_{0}=0) = 1 = y_{0}, \qquad p(x_{1}=1/2) = 9/4 = y_{1}, \qquad p(x_{2}=1) = 4 = y_{2}.

Origin: Stated according to the original source

Problem 4.1.78

(i)(ii)

Let $M$ be the Minkowski space endowed with the standard coordinates $x_{0}, x_{1}, x_{2}, x_{3}$ and the metric tensor field

g = -dx_{0} \otimes dx_{0} + dx_{1} \otimes dx_{1} + dx_{2} \otimes dx_{2} + dx_{3} \otimes dx_{3}

and the quadratic form

q(x) = -(x_{0})^{2} + (x_{1})^{2} + (x_{2})^{2} + (x_{3})^{2}.

Let $\overrightarrow {H}(2)$ be the vector space of $2 \times 2$ hermitian matrices. Consider the map $\varphi : M \rightarrow \overrightarrow {H}(2)$

\varphi (x) = X = \begin{pmatrix} x_{0}+x_{3} & x_{1}-ix_{2} \\ x_{1}+ix_{2} & x_{0}-x_{3} \end{pmatrix}

which is an isomorphism.

Origin: Stated according to the original source

(i)

Find the determinant of $X$ .

(ii)

Find the Cayley transform of $X$ , i.e.

U = (X+iI_{2})^{-1}(X-iI_{2})

with the inverse

X = i(I_{2}+U)(I_{2}-U)^{-1}.

Problem 4.1.79

Let $A$ be an $n \times n$ skew-symmetric matrix over $\mathbb {R}$ . Show that

\operatorname {det}\left(I_{n}+A\right) \geq 1

with equality holding if and only if $A = 0_{n}$ .

Origin: Stated according to the original source

Problem 4.1.80

Consider the $(n-1) \times (n-1)$ symmetric matrix over $\mathbb {R}$

A_{n} = \begin{pmatrix} 3 & 1 & 1 & \ldots & 1 \\ 1 & 4 & 1 & \ldots & 1 \\ 1 & 1 & 5 & \ldots & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \ldots & n+1 \end{pmatrix}

with $n \geq 2$ . Let $D_{n} = \operatorname {det}\left(A_{n}\right)$ . Is the sequence $\left\{ D_{n}/n!\right\}$ bounded?

Origin: Stated according to the original source

Problem 4.1.81

The hyperdeterminant $\operatorname {Det}(A)$ of the three-dimensional array $A = (a_{ijk}) \in \mathbb {R}^{2 \times 2 \times 2}$ can be calculated as follows

\begin{align} \operatorname {Det}(A) & = \frac{1}{4}\left(\operatorname {det}\left(\begin{pmatrix} a_{000} & a_{010} \\ a_{001} & a_{011} \end{pmatrix} + \begin{pmatrix} a_{100} & a_{110} \\ a_{101} & a_{111} \end{pmatrix}\right)\right. \\ & \quad \left. - \operatorname {det}\left(\begin{pmatrix} a_{000} & a_{010} \\ a_{001} & a_{011} \end{pmatrix} - \begin{pmatrix} a_{100} & a_{110} \\ a_{101} & a_{111} \end{pmatrix}\right)\right)^{2} \\ & \quad - 4\operatorname {det}\left(\begin{array}{ll} a_{000} & a_{010} \\ a_{001} & a_{011} \end{array}\right) \operatorname {det}\left(\begin{array}{ll} a_{100} & a_{110} \\ a_{101} & a_{111} \end{array}\right). \end{align}

Assume that only one of the coefficients $a_{ijk}$ is nonzero. Calculate the hyperdeterminant.

Origin: Stated according to the original source

Problem 4.1.82

Let $\epsilon_{00} = \epsilon_{11} = 0$ , $\epsilon_{01} = 1$ , $\epsilon_{10} = -1$ , i.e. we consider the $2 \times 2$ matrix

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

Then the determinant of a $2 \times 2$ matrix $A_{2} = (a_{ij})$ with $i,j = 0,1$ can be defined as

\operatorname {det}\left(A_{2}\right) := \frac{1}{2} \sum _{i=0}^{1} \sum _{j=0}^{1} \sum _{\ell =0}^{1} \sum _{m=0}^{1} \epsilon _{ij}\epsilon _{\ell m}a_{i\ell }a_{jm}.

Thus $\operatorname {det}\left(A_{2}\right) = a_{00}a_{11} - a_{01}a_{10}$ . In analogy the hyperdeterminant $\operatorname {Det}(A_{3})$ of the $2 \times 2 \times 2$ array $A_{3} = (a_{ijk})$ with $i,j,k = 0,1$ is defined as

-\frac{1}{2} \sum _{ii'=0}^{1} \sum _{jj'=0}^{1} \sum _{kk'=0}^{1} \sum _{mm'=0}^{1} \sum _{nn'=0}^{1} \sum _{pp'=0}^{1} \epsilon _{ii'}\epsilon _{jj'}\epsilon _{kk'}\epsilon _{mm'}\epsilon _{nn'}\epsilon _{pp'}a_{ijk}a_{i'j'm}a_{npk'}a_{n'p'm'}.

Calculate $\operatorname {Det}(A_{3})$ .

Origin: Stated according to the original source

Section 4.2Supplementary Problems[43/0%]

Problem 4.2.1

Let $A$ , $B$ be $2 \times 2$ matrices over $\mathbb {C}$ . What are the conditions on $A$ , $B$ such that $\operatorname {tr}(AB) = \operatorname {tr}(A)\operatorname {tr}(B)$ ?

Origin: Stated according to the original source

Problem 4.2.2

Given a $2 \times 2$ matrix $M$ over $\mathbb {C}$ . What can be said about the determinant of $M \otimes I_{2} - I_{2} \otimes M$ ?

Origin: Stated according to the original source

Problem 4.2.3

(i)(ii)

Origin: Stated according to the original source

(i)

What is condition on $\alpha \in \mathbb {R}$ such that

F(\alpha ) = \begin{pmatrix} 1 & \alpha & \alpha \\ \alpha & 1 & \alpha \\ \alpha & \alpha & 1 \end{pmatrix}

is invertible?

(ii)

What is the condition on $\alpha , \beta \in \mathbb {R}$ such that

G(\alpha ,\beta ) = \begin{pmatrix} 1 & \beta & \beta & \beta \\ \beta & 1 & \alpha & \alpha \\ \beta & \alpha & 1 & \alpha \\ \beta & \alpha & \alpha & 1 \end{pmatrix}

is invertible?

Problem 4.2.4

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Then $\operatorname {tr}([A,B]) = 0$ . Consider $n=3$ and

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

with $\operatorname {tr}(C) = 0$ . Construct $3 \times 3$ matrices $A$ and $B$ such that $[A,B] = C$ .

Origin: Stated according to the original source

Problem 4.2.5

Consider the $k$ vectors

\overrightarrow {v}_{j} = \begin{pmatrix} v_{1j} \\ v_{2j} \\ \vdots \\ v_{nj} \end{pmatrix} \qquad j = 1, \ldots , k

in the Euclidean space $\mathbb {R}^{n}$ . The parallelepiped determined by these vectors has the $k$ -dimensional area

\sqrt{\operatorname {det}\left(V^{T}V\right)}

where $V$ is the $n \times k$ matrix with $\overrightarrow {v}_{1}, \overrightarrow {v}_{2}, \ldots , \overrightarrow {v}_{k}$ as its columns. Apply it to the two vectors in $\mathbb {R}^{3}$

\overrightarrow {v}_{1} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \qquad \overrightarrow {v}_{2} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}.

Origin: Stated according to the original source

Problem 4.2.6

The equation of a hyperplane passing through the $n$ points $\overrightarrow {x}_{1}, \overrightarrow {x}_{2}, \ldots , \overrightarrow {x}_{n}$ in the Euclidean space $\mathbb {R}^{n}$ is given by

\operatorname {det}\left(\begin{array}{lllll} 1 & 1 & 1 & \ldots & 1 \\ \overrightarrow {x} & \overrightarrow {x}_{1} & \overrightarrow {x}_{2} & \ldots & \overrightarrow {x}_{n} \end{array}\right) = 0.

Let $n=3$ . Apply it to the three vectors in $\mathbb {R}^{3}$

\overrightarrow {x}_{1} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \qquad \overrightarrow {x}_{2} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \qquad \overrightarrow {x}_{3} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}.

Origin: Stated according to the original source

Problem 4.2.7

Let $A$ , $B$ be $2 \times 2$ matrices. Show that

[A,B]_{+} \equiv AB+BA \equiv (\operatorname {tr}(AB)-\operatorname {tr}(A)\operatorname {tr}(B))I_{2} + \operatorname {tr}(A)B + \operatorname {tr}(B)A.

Can this identity be extended to $3 \times 3$ matrices?

Origin: Stated according to the original source

Problem 4.2.8

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

Origin: Stated according to the original source

(i)

Find all nonzero $2 \times 2$ matrices $A$ and $B$ such that $BA^{*} = \operatorname {tr}(BA^{*})A$ .

(ii)

Find all $2 \times 2$ matrices $A$ such that $A^{2} = \operatorname {tr}(A)A$ . Calculate $\operatorname {det}\left(A\right)$ and $\operatorname {det}\left(A^{2}\right)$ of such a matrix.

(iii)

Let $B$ be a $2 \times 2$ matrix over $\mathbb {C}$ . Assume that $B^{2} = I_{2}$ and thus $\operatorname {tr}(B^{2}) = 2$ . What can be said about the trace of $B$ ?

(iv)

Find all $2 \times 2$ matrices $C$ with $\operatorname {tr}(C^{2})-(\operatorname {tr}(C))^{2} = 0$ .

(v)

Let $D$ be a $2 \times 2$ matrix over $\mathbb {C}$ . Find the condition on $D$ such that $\operatorname {det}\left(D\right) = \operatorname {tr}(D^{2})$ . Find the condition on $D$ such that $\operatorname {det}\left(D \otimes D\right) = \operatorname {tr}(D \otimes D)$ .

Problem 4.2.9

Let $A$ , $B$ be $n \times n$ matrices. Let $U$ , $V$ be unitary $n \times n$ matrices. Show that

\operatorname {tr}((U \otimes V)(A \otimes B)(U^{*} \otimes V^{*})) = \operatorname {tr}(A)\operatorname {tr}(B)

applying cyclic invariance and $UU^{*} = I_{n}$ , $VV^{*} = I_{n}$ .

Origin: Stated according to the original source

Problem 4.2.10

Let $A$ , $B$ be $2 \times 2$ matrices over $\mathbb {C}$ with $\operatorname {tr}(A) = 0$ and $\operatorname {tr}(B) = 0$ . Show that the condition on $A$ and $B$ such that $\operatorname {tr}(AB) = 0$ is given by

2a_{11}b_{11} + a_{12}b_{21} + a_{21}b_{12} = 0.

Origin: Stated according to the original source

Problem 4.2.11

Let $A$ , $B$ be nonzero $2 \times 2$ matrices over $\mathbb {R}$ with $\operatorname {det}\left(A \otimes B\right) = 0$ . Can we conclude that both $A$ and $B$ are noninvertible?

Origin: Stated according to the original source

Problem 4.2.12

Consider the symmetric $n \times n$ band matrix ( $n \geq 3$ )

M_{n} = \begin{pmatrix} 1 & 1 & 0 & \ldots & 0 & 0 \\ 1 & 1 & 1 & \ldots & 0 & 0 \\ 0 & 1 & 1 & \ldots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \ldots & 1 & 1 \\ 0 & 0 & 0 & \ldots & 1 & 1 \end{pmatrix}

with the elements in the field $\mathbb {F}_{2}$ . Show that

\operatorname {det}\left(M_{n}\right) = \operatorname {det}\left(M_{n-1}\right) - \operatorname {det}\left(M_{n-2}\right)

with the initial conditions $\operatorname {det}\left(M_{3}\right) = \operatorname {det}\left(M_{4}\right) = 1$ . Show that the solution is

\operatorname {det}\left(M_{n}\right) = \frac{2\sqrt{3}}{3} \cos \left(\frac{n\pi }{3} - \frac{\pi }{6}\right) \pmod{2}.

Origin: Stated according to the original source

Problem 4.2.13

Consider the nonnormal matrices

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

Find $(\operatorname {det}\left(A_{2}^{*}A_{2}\right))^{1/2}$ and $(\operatorname {det}\left(A_{3}^{*}A_{3}\right))^{1/2}$ . Extend to $n$ dimensions.

Origin: Stated according to the original source

Problem 4.2.14

Let $A$ be a $n \times n$ matrix with $\operatorname {det}\left(A\right) = -1$ . Show that $\operatorname {det}\left(A^{-1}\right) = -1$ .

Origin: Stated according to the original source

Problem 4.2.15

Let $n$ be even. Consider a skew-symmetric $n \times n$ matrix $A$ over $\mathbb {R}$ . Let $B$ be a symmetric $n \times n$ matrix over $\mathbb {R}$ with the entries of $B = (b_{jk})$ given by $b_{jk} = b_{j}b_{k}$ ( $b_{j}, b_{k} \in \mathbb {R}$ ). Show that

\operatorname {det}\left(A+B\right) = \operatorname {det}\left(A\right).

Notice that $\operatorname {det}\left(B\right) = 0$ . Let $S_{n}$ be the symmetric group. Then we have

\operatorname {det}\left(A+B\right) = \sum _{\sigma \in S_{n}} (-1)^{\sigma } (a_{1\sigma (1)}+b_{1}b_{\sigma (1)})(a_{2\sigma (2)}+b_{2}b_{\sigma (2)}) \cdots (a_{n\sigma (n)}+b_{n}b_{\sigma (n)}).

Origin: Stated according to the original source

Problem 4.2.16

Let $h>0$ and $b>a$ . Consider the matrix

M(a,b,h) = \begin{pmatrix} 1 & ah \\ 1 & bh \end{pmatrix}.

Show that $\operatorname {det}\left(M\right) = h(b-a) > 0$ and the inverse of $M$ is given by

M^{-1}(a,b,h) = \begin{pmatrix} b/(b-a) & -a/(b-a) \\ -1/((b-a)h) & 1/((b-a)h) \end{pmatrix}.

Origin: Stated according to the original source

Problem 4.2.17

The $n \times n$ permutation matrices form a group under matrix multiplications. Show that $\operatorname {det}\left(I_{n}-P\right) = 0$ for any $n \times n$ permutation matrix.

Origin: Stated according to the original source

Problem 4.2.18

Let $\overrightarrow {u}, \overrightarrow {v}, \overrightarrow {w}$ be column vectors in $\mathbb {R}^{3}$ . We form the $3 \times 3$ matrix

M = \begin{pmatrix} \overrightarrow {u} & \overrightarrow {v} & \overrightarrow {w} \end{pmatrix}.

Show that $\operatorname {det}\left(M\right) = (\overrightarrow {u} \times \overrightarrow {v}) \cdot \overrightarrow {w}$ .

Origin: Stated according to the original source

Problem 4.2.19

(i)(ii)

Origin: Stated according to the original source

(i)

Let $n \geq 2$ . Consider the $n \times n$ matrix

A = \begin{pmatrix} 1 & 2 & 3 & \ldots & n-1 & n \\ 2 & 3 & 4 & \ldots & n & 1 \\ 3 & 4 & 5 & \ldots & 1 & 2 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ n & 1 & 2 & \ldots & n-2 & n-1 \end{pmatrix}.

Show that $\operatorname {det}\left(A\right) = (-1)^{n(n-1)/2} \frac{1}{2}(n+1)n^{n-1}$ .

(ii)

Let $n \geq 2$ . Consider the $n \times n$ matrix

M(x) = \begin{pmatrix} c_{1} & x & x & \ldots & x & x \\ x & c_{2} & x & \ldots & x & x \\ x & x & c_{3} & \ldots & x & x \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ x & x & x & \ldots & x & c_{n} \end{pmatrix}.

Show that $\operatorname {det}\left(M(x)\right) = (-1)^{n}(P(x)-xdP(x)/dx)$ , where

P(x) = (x-c_{1})(x-c_{2}) \cdots (x-c_{n}).

Problem 4.2.20

Consider the $2 \times 2$ matrix

C = \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix} \; \Rightarrow \; \operatorname {tr}(C) = 1, \; \operatorname {tr}(C^{2}) = 3.

Show that $\operatorname {tr}(C^{k}) = \operatorname {tr}(C^{k-1}) + \operatorname {tr}(C^{k-2})$ with $k=3,4,\ldots$

Origin: Stated according to the original source

Problem 4.2.21

Let $A$ be an $n \times n$ matrix. Assume that

\operatorname {tr}(A^{j}) = 0 \qquad \text{for} \quad j=1,\ldots ,n.

Can we conclude that $\operatorname {det}\left(A\right) = 0$ ?

Origin: Stated according to the original source

Problem 4.2.22

(i)(ii)

Let $A_{1}$ , $A_{2}$ , $A_{3}$ be $2 \times 2$ matrices over $\mathbb {R}$ . We define

c_{j} = \operatorname {tr}(A_{j}), \quad j=1,2,3; \qquad c_{jk} = \operatorname {tr}(A_{j}A_{k}), \quad j,k=1,2,3.

Origin: Stated according to the original source

(i)

Given the coefficients $c_{j}$ , $c_{jk}$ reconstruct the matrices $A_{1}$ , $A_{2}$ , $A_{3}$ .

(ii)

Apply the result to the case $c_{1}=c_{2}=c_{3}=0$

c_{11}=0, \; c_{12}=1, \; c_{13}=0, \; c_{21}=1, \; c_{22}=0, \; c_{23}=0, \; c_{31}=0, \; c_{32}=0, \; c_{33}=2

and show that

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

Problem 4.2.23

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

Origin: Stated according to the original source

(i)

Let $B$ be a $2 \times 2$ matrix over $\mathbb {C}$ . Given

d_{1} = \operatorname {det}\left(B\right), \quad d_{2} = \operatorname {det}\left(B^{2}\right), \quad d_{3} = \operatorname {det}\left(B^{3}\right), \quad d_{4} = \operatorname {det}\left(B^{4}\right).

Can we reconstruct $B$ from $d_{1}, d_{2}, d_{3}, d_{4}$ . Does it depend on whether the matrix $B$ is normal?

(ii)

Find all $2 \times 2$ matrices $A_{1}$ , $A_{2}$ , $A_{3}$ over $\mathbb {C}$ such that

\operatorname {tr}(A_{1}) = \operatorname {tr}(A_{2}) = \operatorname {tr}(A_{3}) = 0, \quad \operatorname {tr}(A_{1}A_{2}) = \operatorname {tr}(A_{2}A_{3}) = \operatorname {tr}(A_{3}A_{1}) = 0.

(iii)

Find all $3 \times 3$ matrices $B_{1}$ , $B_{2}$ , $B_{3}$ over $\mathbb {C}$ such that

\operatorname {tr}(B_{1}) = \operatorname {tr}(B_{2}) = \operatorname {tr}(B_{3}) = 0, \quad \operatorname {tr}(B_{1}B_{2}) = \operatorname {tr}(B_{2}B_{3}) = \operatorname {tr}(B_{3}B_{1}) = 0.

(iv)

Find all $2 \times 2$ matrices $C$ such that $\operatorname {det}\left(C\right) = 1$ , $\operatorname {tr}(C) = 0$ . Do these matrices form a group under matrix multiplication?

Problem 4.2.24

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Assume that $B$ is invertible. Show that $\operatorname {det}\left(I_{n}+BAB^{-1}\right) = \operatorname {det}\left(I_{n}+A\right)$ .

Origin: Stated according to the original source

Problem 4.2.25

An $n \times n$ matrix $A$ is called idempotent if $A^{2} = A$ . Show that $\operatorname {rk}\left(A\right) = \operatorname {tr}(A)$ .

Origin: Stated according to the original source

Problem 4.2.26

Let $A$ be a real symmetric and positive definite $n \times n$ matrix. Then

\ln (\operatorname {det}\left(A\right)) \leq \operatorname {tr}(A) - n.

For the identity matrix $I_{n}$ we obtain equality. Show that for the matrix

A = \frac{1}{2} \begin{pmatrix} 3 & -1 \\ -1 & 3 \end{pmatrix}

we have $n=2$ , $\operatorname {det}\left(A\right) = 2$ , $\operatorname {tr}(A) = 3$ and $\ln (2) < 1$ .

Origin: Stated according to the original source

Problem 4.2.27

Let $A$ , $B$ , $C$ be $n \times n$ matrices over $\mathbb {C}$ . Let $\overrightarrow {a}_{j}$ , $\overrightarrow {b}_{j}$ , $\overrightarrow {c}_{j}$ ( $j=1,\ldots ,n$ ) be the $j$ -th column of $A$ , $B$ , $C$ , respectively. Show that if for some $k \in \left\{ 1,\ldots ,n\right\}$

\overrightarrow {c}_{k} = \overrightarrow {a}_{k} + \overrightarrow {b}_{k}

and

\overrightarrow {c}_{j} = \overrightarrow {a}_{j} = \overrightarrow {b}_{j}, \qquad j=1,\ldots ,k-1,k+1,\ldots ,n

then $\operatorname {det}\left(C\right) = \operatorname {det}\left(A\right)+\operatorname {det}\left(B\right)$ .

Origin: Stated according to the original source

Problem 4.2.28

Consider the Legendre polynomials $p_{j}$ ( $j=0,1,\ldots$ ) with

p_{0}(x) = 1, \qquad p_{1}(x) = x, \qquad p_{2}(x) = \frac{1}{2}(3x^{2}-1),

p_{3}(x) = \frac{1}{2}(5x^{3}-3x), \qquad p_{4}(x) = \frac{1}{8}(35x^{4}-30x^{2}+3).

Show that

\operatorname {det}\left(\begin{array}{lll} p_{0}(x) & p_{1}(x) & p_{2}(x) \\ p_{1}(x) & p_{2}(x) & p_{3}(x) \\ p_{2}(x) & p_{3}(x) & p_{4}(x) \end{array}\right) = (1-x^{2})^{3} \begin{pmatrix} p_{0}(0) & 0 & p_{2}(0) \\ 0 & p_{2}(0) & 0 \\ p_{2}(0) & 0 & p_{4}(0) \end{pmatrix}.

Note that ( $x$ omitted)

\operatorname {det}\left(\begin{array}{lll} p_{0} & p_{1} & p_{2} \\ p_{1} & p_{2} & p_{3} \\ p_{2} & p_{3} & p_{4} \end{array}\right)

= p_{0}(p_{2}p_{4}-(p_{3})^{2}) - p_{1}(p_{1}p_{4}-p_{2}p_{3} + p_{2}(p_{1}p_{3}-(p_{2})^{2})

and $p_{0}(0) = 1$ , $p_{1}(0) = 0$ , $p_{2}(0) = -1/2$ , $p_{3}(0) = 0$ , $p_{4}(0) = 3/8$ .

Origin: Stated according to the original source

Problem 4.2.29

Let $A$ be an $n \times n$ diagonal matrix over $\mathbb {C}$ . Let $B$ be an $n \times n$ matrix over $\mathbb {C}$ with $b_{jj} = 0$ for all $j=1,\ldots ,n$ . Can we conclude that all diagonal elements of the commutator $[A,B]$ are $0$ ?

Origin: Stated according to the original source

Problem 4.2.30

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {R}$ with $\operatorname {det}\left(A\right) = 1$ and $\operatorname {det}\left(B\right) = 1$ . This means $A$ , $B$ are elements of the Lie group $SL(n,\mathbb {R})$ . Can we conclude that

\operatorname {tr}(AB) + \operatorname {tr}(AB^{-1}) = \operatorname {tr}(A)\operatorname {tr}(B)?

Origin: Stated according to the original source

Problem 4.2.31

Let $H$ be a hermitian $n \times n$ matrix. Show that $\operatorname {det}\left(H+iI_{n}\right) \neq 0$ .

Origin: Stated according to the original source

Problem 4.2.32

Let $A$ be an $2 \times 2$ matrix over $\mathbb {R}$ . Calculate $r = \operatorname {tr}(A^{2}) - (\operatorname {tr}(A))^{2}$ . Show that the conditions on $a_{jk}$ such that $r=0$ are given by $a_{12}a_{21} = a_{11}a_{22}$ , i.e. $\operatorname {det}\left(A\right) = 0$ .

Origin: Stated according to the original source

Problem 4.2.33

Consider a triangle embedded in $\mathbb {R}^{3}$ . Let $\overrightarrow {v}_{j} = (x_{j}, y_{j}, z_{j})$ ( $j=1,2,3$ ) be the coordinates of the vertices. Then the area $A$ of the triangle is given by

A = \frac{1}{2}\left\| (\overrightarrow {v}_{2}-\overrightarrow {v}_{1}) \times (\overrightarrow {v}_{1}-\overrightarrow {v}_{3})\right\| = \frac{1}{2}\left\| (\overrightarrow {v}_{3}-\overrightarrow {v}_{1}) \times (\overrightarrow {v}_{3}-\overrightarrow {v}_{2})\right\|

where $\times$ denotes the vector product and $\left\| \cdot \right\|$ denotes the Euclidean norm. The area $A$ of the triangle can also be found via

\frac{1}{2}\sqrt{\left(D\begin{pmatrix} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{pmatrix}\right)^{2} + \left(D\begin{pmatrix} y_{1} & z_{1} & 1 \\ y_{2} & z_{2} & 1 \\ y_{3} & z_{3} & 1 \end{pmatrix}\right)^{2} + \left(D\begin{pmatrix} z_{1} & x_{1} & 1 \\ z_{2} & x_{2} & 1 \\ z_{3} & x_{3} & 1 \end{pmatrix}\right)^{2}}

where $D$ denotes the determinant. Consider

\overrightarrow {v}_{1} = (1,0,0), \qquad \overrightarrow {v}_{2} = (0,1,0), \qquad \overrightarrow {v}_{3} = (0,0,1).

Find the area of the triangle using both expressions. Discuss. The triangle could be one of the faces of a tetrahedron.

Origin: Stated according to the original source

Problem 4.2.34

Consider the $3 \times 3$ matrix $M$ with entries

(M)_{jk} = x_{k}^{j-1}, \qquad j,k = 1,2,3.

Find the determinant of this matrix.

Origin: Stated according to the original source

Problem 4.2.35

The permanent and the determinant of an $n \times n$ matrix $M$ over $\mathbb {C}$ are respectively defined as

\operatorname {perm}(M) := \sum _{\pi \in S_{n}} \left(\prod _{j=1}^{n} M_{j,\pi (j)}\right), \qquad \operatorname {det}\left(M\right) := \sum _{\pi \in S_{n}} (-1)^{\operatorname {sgn}(\pi )} \left(\prod _{j=1}^{n} M_{j,\pi (j)}\right)

where $S_{n}$ denotes the symmetric group on a set of $n$ symbols. For an $n \times n$ matrix $A$ and an $m \times m$ matrix $B$ we know that

\operatorname {det}\left(A \otimes B\right) \equiv (\operatorname {det}\left(A\right))^{m}(\operatorname {det}\left(B\right))^{n}.

Study $\operatorname {perm}(A \otimes B)$ .

Origin: Stated according to the original source

Problem 4.2.36

(i)(ii)

Origin: Stated according to the original source

(i)

Let $A$ , $B$ be positive definite $n \times n$ matrices. Show that $\operatorname {tr}(A+B) \geq \operatorname {tr}(A)$ .

(ii)

Let $C$ , $D$ be $n \times n$ matrices over $\mathbb {C}$ . Is $\operatorname {tr}(CC^{*}DD^{*}) \leq \operatorname {tr}(CC^{*})\operatorname {tr}(DD^{*})$ ?

Problem 4.2.37

Let $A$ , $B$ , $C$ , $D$ be $n \times n$ matrices. Assume that $D$ is invertible. Consider the $(2n) \times (2n)$ matrix

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

Show that $\operatorname {det}\left(M\right) = \operatorname {det}\left(AD-BD^{-1}CD\right)$ using the identity

\begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} I_{n} & 0_{n} \\ -D^{-1}C & I_{n} \end{pmatrix} \equiv \begin{pmatrix} A-BD^{-1}C & B \\ 0_{n} & D \end{pmatrix}.

Origin: Stated according to the original source

Problem 4.2.38

Let $\mu \in \mathbb {R}$ and $A$ be an $2 \times 2$ matrix over $\mathbb {R}$ . Show that the determinant of the $4 \times 4$ matrix

\begin{pmatrix} -\mu I_{2} & A \\ A^{T} & -\mu I_{2} \end{pmatrix}

only contains even powers of $\mu$ .

Origin: Stated according to the original source

Problem 4.2.39

Consider the invertible $3 \times 3$ matrix $A$

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

Find a permutation matrix $P$ such that $PA^{-1}P^{-1} = A$ .

Origin: Stated according to the original source

Problem 4.2.40

Let $x, \epsilon \in \mathbb {R}$ . Show that the determinant of the symmetric $n \times n$ matrix

A(x,\epsilon ) = \begin{pmatrix} x+\epsilon & x & & x \\ x & x+\epsilon & & x \\ & & \ddots & \\ x & x & & x+\epsilon \end{pmatrix}

is given by $\operatorname {det}\left(A(x,\epsilon )\right) = \epsilon^{n-1}(nx+\epsilon )$ .

Origin: Stated according to the original source

Problem 4.2.41

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

\Omega = \begin{pmatrix} 0_{n} & I_{n} \\ -I_{n} & 0_{n} \end{pmatrix} \equiv \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \otimes I_{n}

is given by $\operatorname {det}\left(\Omega \right) = \left(\operatorname {det}\left(\begin{array}{ll} 0 & 1 \\ -1 & 0 \end{array}\right)\right)^{n} (\operatorname {det}\left(I_{n}\right))^{2} = 1$ .

Origin: Stated according to the original source

Problem 4.2.42

Given a $2 \times 2 \times 2$ hypermatrix $A = (a_{jk\ell })$ , $j,k,\ell = 0,1$ and the $2 \times 2$ matrix

S = \begin{pmatrix} s_{00} & s_{01} \\ s_{10} & s_{11} \end{pmatrix}.

The multiplication $AS$ which is again a $2 \times 2$ hypermatrix is defined by

(AS)_{jk\ell } := \sum _{r=0}^{1} a_{jkr}s_{r\ell }.

Assume that $\operatorname {det}\left(S\right) = 1$ , i.e. $S \in SL(2,\mathbb {C})$ . Show that $\operatorname {Det}(AS) = \operatorname {Det}(A)$ . This is a typical problem to apply computer algebra.

Origin: Stated according to the original source

Problem 4.2.43

The Levi-Civita symbol (also called completely antisymmetric constant tensor) is defined by

\epsilon _{j_{1},j_{2},\ldots ,j_{n}} := \begin{cases} +1, & \text{if } j_{1},j_{2},\ldots ,j_{n} \text{ is an even permutation of } 12\ldots n \\ -1, & \text{if } j_{1},j_{2},\ldots ,j_{n} \text{ is an odd permutation of } 12\ldots n \\ 0, & \text{otherwise} \end{cases}

Let $\delta_{jk}$ be the Kronecker delta. Show that

\epsilon _{j_{1},j_{2},\ldots ,j_{n}} \epsilon _{k_{1},k_{2},\ldots ,k_{n}} = \operatorname {det}\left(\begin{array}{llll} \delta _{j_{1}k_{1}} & \delta _{j_{2}k_{1}} & & \delta _{j_{n}k_{1}} \\ \delta _{j_{1}k_{2}} & \delta _{j_{2}k_{2}} & & \delta _{j_{n}k_{2}} \\ \vdots & \vdots & & \vdots \\ \delta _{j_{1}k_{n}} & \delta _{j_{2}k_{n}} & \ldots & \delta _{j_{n}k_{n}} \end{array}\right).

Origin: Stated according to the original source