Math Problems Vault

Problem 15.66

Let $A, B$ be diagonal matrices of the same order, $\alpha$ a number. Prove that the matrices $\alpha A, A + B, AB, BA$ are also diagonal and $AB = BA$ .

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Problem 15.67

(1)(2)

Let $A = \operatorname {diag}\left(\lambda_{1}, \ldots , \lambda_{n}\right)$ . Prove that:

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(1)

the columns of the matrix $BA$ are obtained by multiplying the columns of the matrix $B$ by the numbers $\lambda_{1}, \ldots , \lambda_{n}$

(2)

the rows of the matrix $AB$ are obtained by multiplying the rows of $B$ by the numbers $\lambda_{1}, \ldots , \lambda_{n}$ .

Problem 15.68

Let $A$ be a diagonal matrix, $f(t)$ a polynomial. Prove that then the matrix $f(A)$ is also diagonal.

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Problem 15.69

Let the matrix $A$ be diagonal, all its diagonal elements distinct, and $AB = BA$ . Prove that then the matrix $B$ is also diagonal.

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Problem 15.70

The matrix $A$ commutes with every diagonal matrix of order $n$ . Prove that $A$ is a diagonal matrix of order $n$ .

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Problem 15.71

The matrix $A$ commutes with all matrix units of order $n$ . Prove that $A$ is a scalar matrix.

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Problem 15.72

The matrix $A$ commutes with every matrix of order $n$ . Prove that $A$ is a scalar matrix.

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Problem 15.73

Find all matrices that commute with every nonsingular matrix of order $n$ .

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Problem 15.74

(1)(2)(3)(4)

Find the matrix Hermitian conjugate to the given matrix:

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(1)

$\mathbf{A}_{82}$ . $\mathbf{A}_{82} = \left[\begin{smallmatrix} i & 1 \\ -1 & i \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{86}$ . $\mathbf{A}_{86} = \left[\begin{smallmatrix} 5 & i \\ -i & 1 \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{89}$ . $\mathbf{A}_{89} = \left[\begin{smallmatrix} 1 & -i \\ 2+i & 1-2 i \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{81}$ . $\mathbf{A}_{81} = \left[\begin{smallmatrix} 1+i \sqrt{2} & 3 \\ 1 & 1-i \sqrt{2} \end{smallmatrix}\right]$ .

Problem 15.75

Verify the validity of the identity:

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(1)

$(A + B)^{H} = A^{H} + B^{H}$

(2)

$(\alpha A)^{H} = \bar{\alpha } A^{H}$

(3)

$\left(A^{H}\right)^{H} = A$

(4)

$(A \cdot B)^{H} = B^{H} A^{H}$

(5)

$\left(A^{H}\right)^{ - 1} = \left(A^{ - 1}\right)^{H}$ .

Problem 15.76

Determine whether the given second-order matrix is diagonal, scalar, triangular, symmetric, skew-symmetric, Hermitian, skew-Hermitian, unitary, orthogonal, or a permutation matrix:

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(1)

$\mathbf{A}_{82}$ . $\mathbf{A}_{82} = \left[\begin{smallmatrix} i & 1 \\ -1 & i \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{12}$ . $\mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{87}$ . $\mathbf{A}_{87} = \left[\begin{smallmatrix} 1 & 1+i \\ 1-i & 3 \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{86}$ . $\mathbf{A}_{86} = \left[\begin{smallmatrix} 5 & i \\ -i & 1 \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{77}$ . $\mathbf{A}_{77} = \left[\begin{smallmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{smallmatrix}\right]$ .

(6)

$\mathbf{A}_{15}$ . $\mathbf{A}_{15} = \left[\begin{smallmatrix} -1 & 0 \\ 0 & 1 \end{smallmatrix}\right]$ .

(7)

$\mathbf{A}_{46}$ . $\mathbf{A}_{46} = \left[\begin{smallmatrix} -4 & 0 \\ 1 & 4 \end{smallmatrix}\right]$ .

(8)

$\mathbf{A}_{23}$ . $\mathbf{A}_{23} = \left[\begin{smallmatrix} 0 & 2 \\ -2 & 0 \end{smallmatrix}\right]$ .

(9)

$\frac{1}{\sqrt{2}} \mathbf{A}_{88}$ . $\mathbf{A}_{88} = \left[\begin{smallmatrix} 1 & i \\ i & 1 \end{smallmatrix}\right]$ .

(10)

$\mathbf{A}_{22}$ . $\mathbf{A}_{22} = \left[\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\right]$ .

(11)

$\frac{1}{\sqrt{2}} \mathbf{A}_{16}$ . $\mathbf{A}_{16} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & 1 \end{smallmatrix}\right]$ .

Problem 15.77

(1)(2)(3)

Prove that:

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(1)

all diagonal elements of a skew-symmetric matrix equal 0

(2)

the diagonal elements of a Hermitian matrix are real

(3)

the diagonal elements of a skew-Hermitian matrix are imaginary.

Problem 15.78

(1)(2)

Prove that:

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(1)

if the matrix $A$ is Hermitian, then the matrix $iA$ is skew-Hermitian

(2)

if the matrix $A$ is skew-Hermitian, then $iA$ is Hermitian.

Problem 15.79

(1)(2)(3)

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(1)

Find the general form of Hermitian matrices of the second order.

(2)

Find the general form of skew-Hermitian matrices of the second order.

(3)

Indicate all permutation matrices of the second order.

Problem 15.80

Prove: if the matrix $A$ is diagonal and all its diagonal elements are nonzero, then $A^{ - 1}$ exists and is diagonal.

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Problem 15.81

Prove: if the matrix $A$ is upper triangular and all its diagonal elements are nonzero, then $A^{ - 1}$ exists and is upper triangular.

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Problem 15.82

Prove: if $A$ is a nonsingular symmetric matrix, then $A^{ - 1}$ is also a symmetric matrix.

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Problem 15.83

Prove: if $A$ is a nonsingular skew-symmetric matrix, then $A^{ - 1}$ is also a skew-symmetric matrix.

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Problem 15.84

Prove: if $A$ is an orthogonal matrix, then $A^{ - 1}$ exists and is orthogonal.

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Problem 15.85

Prove: if $A$ is a unitary matrix, then $A^{ - 1}$ exists and is unitary.

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Problem 15.86

Prove: if $A$ is a permutation matrix, then $A^{ - 1}$ exists and is also a permutation matrix.

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Problem 15.87

Prove that the given matrix is orthogonal and find its inverse:

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(1)

$\mathbf{A}_{77}$ . $\mathbf{A}_{77} = \left[\begin{smallmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{313}$ . $\mathbf{A}_{313} = \left[\begin{smallmatrix} 2/3 & 1/\sqrt{2} & -1/(3\sqrt{2}) \\ 1/3 & 0 & 4/(3\sqrt{2}) \\ -2/3 & 1/\sqrt{2} & 1/(3\sqrt{2}) \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{318}$ . $\mathbf{A}_{318} = \left[\begin{smallmatrix} 1/\sqrt{10} & 1/\sqrt{2} & 2/\sqrt{10} \\ 1/\sqrt{10} & -1/\sqrt{2} & 2/\sqrt{10} \\ -2/\sqrt{5} & 0 & 1/\sqrt{5} \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{322}$ . $\mathbf{A}_{322} = \left[\begin{smallmatrix} 1/\sqrt{2} & 1/\sqrt{3} & 1/\sqrt{6} \\ 0 & 1/\sqrt{3} & -2/\sqrt{6} \\ -1/\sqrt{2} & 1/\sqrt{3} & 1/\sqrt{6} \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{445}$ . $\mathbf{A}_{445} = \left[\begin{smallmatrix} 1/\sqrt{10} & 1/2 & 1/2 & 2/\sqrt{10} \\ 2/\sqrt{10} & 1/2 & -1/2 & -1/\sqrt{10} \\ 1/\sqrt{10} & -1/2 & -1/2 & 2/\sqrt{10} \\ 2/\sqrt{10} & -1/2 & 1/2 & -1/\sqrt{10} \end{smallmatrix}\right]$ .

Problem 15.88

(1)(2)

Prove that the given matrix is unitary and find its inverse:

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(1)

$\mathbf{A}_{103}$ . $\mathbf{A}_{103} = \left[\begin{smallmatrix} 0 & i \\ -i & 0 \end{smallmatrix}\right]$ .

(2)

$\frac{1}{\sqrt{2}} \mathbf{A}_{488}$ . $\mathbf{A}_{488} = \left[\begin{smallmatrix} i & 0 & -i & 0 \\ 0 & i & 0 & -i \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{smallmatrix}\right]$ .

Problem 15.89

Let the matrices $A$ and $B$ be upper triangular. Express the elements of the matrix $AB$ in terms of the elements of the matrices $A$ and $B$ .

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Problem 15.90

Let the matrices $A$ and $B$ be upper triangular. Prove that the matrices $A + B$ and $AB$ are also upper triangular.

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Problem 15.91

(1)(2)(3)

Let the matrices $A$ and $B$ be symmetric. Prove that:

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(1)

$A + B$ is a symmetric matrix

(2)

$A^{k}$ is a symmetric matrix for any natural number $k$

(3)

the matrix $AB$ is symmetric if and only if the matrices $A$ and $B$ commute.

Problem 15.92

(1)(2)(3)(4)

Let the matrices $A$ and $B$ be skew-symmetric. Prove that:

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(1)

$A + B$ is a skew-symmetric matrix

(2)

$A^{k}$ is a skew-symmetric matrix for odd $k$ and a symmetric matrix for even $k$

(3)

the matrix $AB$ is symmetric if and only if $A$ and $B$ commute.

(4)

State and prove a necessary and sufficient condition for the skew-symmetry of the product of the matrices $A$ and $B$ .

Problem 15.93

Let $A$ be an arbitrary square matrix. Prove that the matrices $A + A^{T}$ and $AA^{T}$ are symmetric, and the matrix $A - A^{T}$ is skew-symmetric.

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Problem 15.94

Prove that any square matrix can be decomposed into the sum of a symmetric and a skew-symmetric matrix. Is this decomposition unique?

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Problem 15.95

(1)(2)(3)

Decompose the given matrix into the sum of a symmetric and a skew-symmetric matrix:

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(1)

$\mathbf{A}_{49}$ . $\mathbf{A}_{49} = \left[\begin{smallmatrix} 4 & -3 \\ 12 & -8 \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{16}$ . $\mathbf{A}_{16} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & 1 \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{234}$ . $\mathbf{A}_{234} = \left[\begin{smallmatrix} 0 & 2 & 2 \\ 0 & -1 & -2 \\ 0 & 0 & -2 \end{smallmatrix}\right]$ .

Problem 15.96

Let $S$ be a nonsingular matrix and $S^{T} AS = B$ . Prove that each of the properties: symmetry, skew-symmetry — holds for the matrices $A$ and $B$ simultaneously (i.e., if it holds for the matrix $A$ , then it holds for $B$ as well, and conversely).

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Problem 15.97

Prove the statement: every real Hermitian matrix is symmetric.

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Problem 15.98

(1)(2)

Let the matrices $A$ and $B$ be Hermitian. Prove that:

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(1)

the matrix $A + B$ is Hermitian

(2)

the matrix $AB$ is Hermitian if and only if the matrices $A$ and $B$ commute.

Problem 15.99

Let $A$ be a Hermitian matrix and $A = B + iC$ , where $B$ and $C$ are real matrices. Prove that $B$ is a symmetric matrix, and $C$ is skew-symmetric.

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Problem 15.100

Prove that any square matrix can be decomposed into the sum of a Hermitian and a skew-Hermitian matrix. Is this decomposition unique?

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Problem 15.101

Prove: a real unitary matrix is orthogonal.

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Problem 15.102

Prove: if the matrices $A$ and $B$ are orthogonal, then $AB$ is orthogonal.

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Problem 15.103

Prove: if the matrices $A$ and $B$ are unitary, then $AB$ is unitary.

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Problem 15.104

Let $A$ be an orthogonal matrix. Prove that then the sum of the squares of the elements of any of its rows equals 1, and the sum of the pairwise products of the corresponding elements of two different rows equals 0. Are these properties defining?

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Problem 15.105

State and prove the properties of the columns of an orthogonal matrix, analogous to 15. 104.

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Problem 15.106

State and prove the properties of a unitary matrix, analogous to properties $15. 104, 15. 105$ of an orthogonal matrix.

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Problem 15.107

Prove that a permutation matrix is orthogonal.

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Problem 15.108

Prove that if $A$ and $B$ are permutation matrices, then $AB$ is also a permutation matrix.

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Problem 15.109

It is known that the matrix $A$ is diagonal and orthogonal. What can be said about its diagonal elements $\lambda_{i}$ ?

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Problem 15.110

The matrix $A$ is diagonal and unitary. What can be said about its diagonal elements $\lambda_{i}$ ?

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Problem 15.111

Verify whether the given matrix is periodic, nilpotent, or stochastic; find the period, the index of nilpotency:

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(1)

$\mathbf{A}_{22}$ . $\mathbf{A}_{22} = \left[\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{14}$ . $\mathbf{A}_{14} = \left[\begin{smallmatrix} 1 & 0 \\ 1 & 0 \end{smallmatrix}\right]$ .

(3)

$\frac{1}{2} \mathbf{A}_{12}$ . $\mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{5}$ . $\mathbf{A}_{5} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & -1 \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{77}$ . $\mathbf{A}_{77} = \left[\begin{smallmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{smallmatrix}\right]$ .

(6)

$\mathbf{A}_{243}$ . $\mathbf{A}_{243} = \left[\begin{smallmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{smallmatrix}\right]$ .

(7)

$\mathbf{A}_{235}$ . $\mathbf{A}_{235} = \left[\begin{smallmatrix} 1 & 2 & 1 \\ 1 & 2 & 4 \\ -1 & -2 & -3 \end{smallmatrix}\right]$ .

(8)

$\mathbf{A}_{237}$ . $\mathbf{A}_{237} = \left[\begin{smallmatrix} -1 & 1 & -2 \\ 3 & -3 & 6 \\ 2 & -2 & 4 \end{smallmatrix}\right]$ .

(9)

$\mathbf{A}_{236}$ . $\mathbf{A}_{236} = \left[\begin{smallmatrix} -2 & 5 & 3 \\ -2 & 5 & 3 \\ 2 & -5 & -3 \end{smallmatrix}\right]$ .

(10)

$\mathbf{A}_{430}$ . $\mathbf{A}_{430} = \left[\begin{smallmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{smallmatrix}\right]$ .

(11)

$\mathbf{A}_{431}$ . $\mathbf{A}_{431} = \left[\begin{smallmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{smallmatrix}\right]$ .

(12)

$\mathbf{A}_{457}$ . $\mathbf{A}_{457} = \left[\begin{smallmatrix} 1 & 0 & 0 & -1 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & -1 & 0 \\ 0 & 1 & -1 & 0 \end{smallmatrix}\right]$ .

(13)

$\frac{1}{7} \mathbf{A}_{434}$ . $\mathbf{A}_{434} = \left[\begin{smallmatrix} 1 & 2 & 2 & 2 \\ 2 & 1 & 2 & 2 \\ 2 & 2 & 1 & 2 \\ 2 & 2 & 2 & 1 \end{smallmatrix}\right]$ .

(14)

$\mathbf{A}_{613}$ . $\mathbf{A}_{613} = \left[\begin{smallmatrix} 0 & 1 & 0 & \ldots & 0 & 0 \\ 0 & 0 & 1 & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 0 & 1 \\ 0 & 0 & 0 & \ldots & 0 & 0 \end{smallmatrix}\right]$ .

Problem 15.112

Verify the property: a nilpotent matrix is always singular, a periodic one is nonsingular.

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Problem 15.113

Verify the property: if $A$ is a nilpotent matrix of second order, then $A^{2} = O$ .

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Problem 15.114

Verify the property: a triangular matrix is nilpotent if and only if all its diagonal elements are zero.

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Problem 15.115

Verify the property: if the matrices $A, B$ are nilpotent and commute, then $A + B$ and $AB$ are nilpotent.

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Problem 15.116

Verify the property: if the matrices $A$ and $B$ are periodic and commute, then $AB$ is a periodic matrix. Express some period of it in terms of the periods of the matrices $A, B$ .

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Problem 15.117

Verify the property: let $A^{m} + A^{m - 1} + \ldots + E = O$ . Prove that $A$ is a periodic matrix.

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Problem 15.118

Verify the property: every permutation matrix is periodic.

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Problem 15.119

Verify the property: let the matrix $A$ be simultaneously unitary and Hermitian. Prove that $A$ is periodic.

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Problem 15.120

Let $S$ be a nonsingular matrix and $S^{ - 1} AS = B$ . Verify that each of the properties: periodicity, nilpotency — holds for the matrices $A$ and $B$ simultaneously (i.e., if it holds for the matrix $A$ , then it holds for $B$ as well, and conversely).

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Problem 15.121

Let the matrices $A$ and $B$ be nonnegative. Verify that then $A + B, AB$ are also nonnegative matrices.

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Problem 15.122

Let $I$ be a column of ones, and let the matrix $A$ be nonnegative. Prove that the condition $AI = I$ is a necessary and sufficient condition for $A$ to be stochastic.

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Problem 15.123

Prove that if the matrices $A$ and $B$ are stochastic, then the matrix $AB$ is also stochastic.

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Problem 15.124

Let the matrix $A$ be stochastic. Does $A^{ - 1}$ exist? Will $A^{ - 1}$ be stochastic, if it exists?

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Problem 15.125

In what case is a stochastic matrix orthogonal?

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Problem 15.126

(1)(2)

Prove the validity of the identity:

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(1)

$\operatorname {tr}(A + B) = \operatorname {tr} A + \operatorname {tr} B$

(2)

$\operatorname {tr} AB = \operatorname {tr} BA$ .

Problem 15.127

Let $A$ be a triangular matrix, $m$ a natural number. Compute the trace of the matrix $A^{m}$ .

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Problem 15.128

(1)(2)(3)

Let $A$ be an arbitrary matrix. Compute:

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(1)

$\operatorname {tr}\left(A^{T} A\right)$

(2)

$\operatorname {tr}\left(A^{H} A\right)$

(3)

Prove that if $\operatorname {tr}\left(A^{H} A\right) = 0$ , then $A = O$ .

Problem 15.129

Prove that if $A$ is a nilpotent matrix of second order, then $\operatorname {tr} A = 0$ .

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Problem 15.130

Prove that there do not exist matrices $A$ and $B$ such that $AB - BA = E$ .

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