Math Problems Vault

Problem 24.1

Prove that the set of all eigenvectors of a linear transformation corresponding to one eigenvalue, augmented with the zero vector, coincides with the eigenspace.

?

Problem 24.2

(1)(2)

Prove that:

?

(1)

the kernel of a linear transformation coincides with the eigenspace corresponding to the zero eigenvalue

(2)

The eigenspace of the transformation $\varphi$ corresponding to the eigenvalue $\lambda$ is the set of vectors satisfying the condition $\varphi (x) = \lambda x$ .

Problem 24.3

Let $A$ be the matrix, and $\lambda$ an eigenvalue, of a linear transformation of an $n$ -dimensional vector space. What is the dimension of the eigenspace corresponding to the eigenvalue $\lambda$ , if the rank of the matrix $A - \lambda E$ equals $r$ ?

?

Problem 24.4

What form does the matrix of a linear transformation have, if the first $k$ basis vectors are its eigenvectors?

?

Problem 24.5

Prove that the dimension of the eigenspace corresponding to a given root of the characteristic polynomial does not exceed the multiplicity of this root.

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Problem 24.6

Let $x, y$ be eigenvectors of a linear transformation corresponding to different eigenvalues, and let the numbers $\alpha , \beta$ be different from zero. Prove that the vector $\alpha x + \beta y$ is not an eigenvector.

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Problem 24.7

Prove that a nonzero linear transformation for which all nonzero vectors are eigenvectors is a homothety.

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Problem 24.8

Prove that a system of eigenvectors corresponding to pairwise distinct eigenvalues of a linear transformation is linearly independent.

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Problem 24.9

Prove that the matrix of a linear transformation in some basis is diagonal if and only if all the vectors of the basis are eigenvectors.

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Problem 24.10

Prove that a linear transformation of an $n$ -dimensional vector space having $n$ distinct eigenvalues is diagonalizable.

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Problem 24.11

Let $\varphi$ be a linear transformation of a finite-dimensional vector space $\mathcal{L}$ . Prove that the following statements are equivalent:

?

(1)

$\varphi$ is diagonalizable

(2)

there exists in $\mathcal{L}$ a basis of eigenvectors of the transformation $\varphi$

(3)

the union of the bases of the eigenspaces is a basis in $\mathcal{L}$

(4)

the multiplicity of each root $\lambda$ of the characteristic equation equals the dimension of the eigenspace $\mathcal{L}_{\lambda }$

(5)

$\mathcal{L}$ is the direct sum of the eigenspaces.

Problem 24.12

(1)(2)

Prove that:

?

(1)

in a complex vector space every characteristic number of the matrix of a linear transformation is an eigenvalue, so that an arbitrary linear transformation has at least one eigenvector

(2)

in a real space every real characteristic number is an eigenvalue.

Problem 24.13

Prove that a linear transformation of an odd-dimensional (for example, three-dimensional) real vector space has at least one eigenvector.

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Problem 24.14

(1)(2)

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(1)

Prove that the characteristic polynomial, the determinant, and the trace of the matrix of a linear transformation do not depend on the choice of basis.

(2)

Find an expression for the coefficients of the characteristic polynomial, in particular the trace and the determinant of a matrix of order $n$ , in terms of the characteristic numbers.

Problem 24.15

(1)(2)(3)

Find the eigenvectors and eigenvalues of each of the following transformations:

?

(1)

the zero one

(2)

the identity one

(3)

a homothety.

Problem 24.16

Find the eigenvalues and eigenvectors of a linear transformation given, in some basis of an $n$ -dimensional vector space, by the matrix $\mathbf{A}_{612} = J_{n}(\lambda )$ . $\mathbf{A}_{612} = \left[\begin{smallmatrix} \lambda & 1 & 0 & \ldots & 0 \\ 0 & \lambda & 1 & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 1 \\ 0 & 0 & 0 & \ldots & \lambda \end{smallmatrix}\right]$ .

?

Problem 24.17

Let the matrix of a linear transformation in some basis be upper or lower triangular with diagonal entries $\lambda_{1}, \ldots , \lambda_{n}$ . Find all eigenvalues of this transformation.

?

Problem 24.18

(1)(2)

Let $\mathcal{L} = \mathcal{L}^{\prime } \oplus \mathcal{L}^{\prime \prime }$ , where $\mathcal{L}^{\prime }, \mathcal{L}^{\prime \prime }$ are nonzero subspaces. Find the eigenvalues and eigenspaces of the linear transformation $\varphi$ ; prove that $\varphi$ has a basis of eigenvectors, and indicate the diagonal form of its matrix, if $\varphi$ is:

?

(1)

projection onto the subspace $\mathcal{L}^{\prime }$ parallel to $\mathcal{L}^{\prime \prime }$

(2)

reflection in the subspace $\mathcal{L}^{\prime }$ parallel to $\mathcal{L}^{\prime \prime }$ .

Problem 24.19

(1)(2)(3)

Find the eigenvalues and eigenspaces, reduce to diagonal form the matrices of the linear transformations defined in problem 23.65.

?

(1)

reflection of the plane in the line $x + 2y = 0$ parallel to the line $x + 3y = 0$

(2)

projection of the plane onto the line $x + y = 0$ parallel to the line $4x + 5y = 0$

(3)

contraction with coefficient $\lambda = 2$ toward the line $3x - 2y = 0$ parallel to the line $x - y = 0$ .

Problem 24.20

Find the eigenvalues, eigenspaces, reduce to diagonal form the matrix of the linear transformation defined in problem:

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(1)

23.9(1): orthogonal projection of the space $\mathcal{E}_{3}$ onto the subspace $\mathcal{L}$ , if $\mathcal{L}$ is the line $x = z = 0$ ;

(2)

23.9(2): orthogonal projection of the space $\mathcal{E}_{3}$ onto the subspace $\mathcal{L}$ , if $\mathcal{L}$ is the line $x = y = z$ ;

(3)

23.9(3): orthogonal projection of the space $\mathcal{E}_{3}$ onto the subspace $\mathcal{L}$ , if $\mathcal{L}$ is the plane $x + y + z = 0$ ;

(4)

23.9(4): orthogonal projection of the space $\mathcal{E}_{3}$ onto the subspace $\mathcal{L}$ , if $\mathcal{L}$ is the plane spanned by the vectors $\mathbf{a}(- 1, 1, - 1)$ and $\mathbf{b}(1, - 3, 2)$ ;

(5)

23.10(1): projection of the space $\mathcal{E}_{3}$ onto the subspace $\mathcal{L}$ parallel to the subspace $\mathcal{M}$ , if: $\mathcal{L}$ is defined by the equation $x = 0$ , $\mathcal{M}$ by the equations $2x = 2y = - z$ ;

(6)

23.10(2): projection of the space $\mathcal{E}_{3}$ onto the subspace $\mathcal{L}$ parallel to the subspace $\mathcal{M}$ , if: $\mathcal{L}$ has the equation $x = y$ , $\mathcal{M}$ is defined by the system of equations $x + y + z = 0, 2x + y + 4z = 0$ ;

(7)

23.10(3): projection of the space $\mathcal{E}_{3}$ onto the subspace $\mathcal{L}$ parallel to the subspace $\mathcal{M}$ , if: $\mathcal{L}$ is defined by the equations $- 20x = 15y = 12z$ , $\mathcal{M}$ by the equation $2x + 3y - z = 0$ ;

(8)

23.10(4): projection of the space $\mathcal{E}_{3}$ onto the subspace $\mathcal{L}$ parallel to the subspace $\mathcal{M}$ , if: $\mathcal{L}$ is defined by the system of equations $x - y + z = 0, 2x - 3y + 4z = 0$ , $\mathcal{M}$ by the equation $2x + 3y - 4z = 0$ ;

(9)

23.12(1): the linear transformation $\varphi$ , if $\varphi$ is the orthogonal reflection of the space $\mathcal{E}_{3}$ with respect to the plane $x = 0$ ;

(10)

23.12(2): the linear transformation $\varphi$ , if $\varphi$ is the orthogonal reflection of the space $\mathcal{E}_{3}$ with respect to the line $x = 2y = z$ ;

(11)

23.12(3): the linear transformation $\varphi$ , if $\varphi$ is the orthogonal reflection of the space $\mathcal{E}_{3}$ with respect to the linear span of the vectors $\mathbf{a}(1, 0, - 1)$ and $\mathbf{b}(1, 1, - 2)$ ;

(12)

23.13(1): reflection of the space $\mathcal{E}_{3}$ in the plane $x = 0$ parallel to the line $2x = y = - z$ ;

(13)

23.13(2): reflection of the space $\mathcal{E}_{3}$ in the line $x = z, \quad x - y + z = 0$ parallel to the plane $x + y = 0$ .

The vector subspaces of the three-dimensional geometric vector space $\mathcal{E}_{3}$ are given by their equations in some orthonormal basis.

Problem 24.21

Find the eigenvalues and eigenvectors of the linear transformation defined in problem:

?

(1)

$23. 14, 1)$

(2)

$23. 14, 2)$

(3)

$23. 14, 3)$

(4)

$23. 66, 1)$

(5)

$23. 66, 2)$

(6)

$23. 66, 3)$

(7)

$23. 66, 4)$

(8)

$23. 66, 5)$ .

Is it possible to form a basis from the eigenvectors of the transformation?

Problem 24.22

(1)(2)(3)

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(1)

Find the eigenvalues and eigenvectors of the linear transformation $\varphi$ given by the matrix

A = \left(a_{1}, \ldots , a_{n}\right)^{T}\left(b_{1}, \ldots , b_{n}\right) \neq 0.

(2)

Find a necessary and sufficient condition for the diagonalizability of the transformation $\varphi$ .

(3)

Determine whether the transformations given by the matrices are diagonalizable:

(1) $\mathbf{A}_{213}$ ;

(2) $\mathbf{A}_{222}$ . $\mathbf{A}_{213} = \left[\begin{smallmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 3 & 3 & 3 \end{smallmatrix}\right], \mathbf{A}_{222} = \left[\begin{smallmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \\ -3 & -3 & -3 \end{smallmatrix}\right]$ .

Problem 24.23

Let $k, m, n$ be natural numbers, $1 \leqslant k \leqslant m \leqslant n$ . Give an example of a linear transformation of an $n$ -dimensional vector space for which a given number $\lambda$ is a root of the characteristic polynomial of multiplicity $m$ , and the eigenspace corresponding to it has dimension $k$ .

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Problem 24.24

Let a linear transformation $\varphi$ of a three-dimensional complex vector space have, in some basis, a real matrix, and let at least one characteristic number of this matrix not be real. Prove that $\varphi$ is diagonalizable.

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Problem 24.25

Let $x$ be an eigenvector of a linear transformation $\varphi$ corresponding to the eigenvalue $\lambda$ , and let $p(t)$ be a polynomial. Prove that the vector $x$ is an eigenvector for the transformation $p(\varphi )$ and belongs to the eigenvalue $p(\lambda )$ .

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Problem 24.26

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

Let $\lambda_{1}, \ldots , \lambda_{n}$ be the characteristic numbers of a linear transformation $\varphi$ in an $n$ -dimensional vector space. What are the characteristic numbers (with multiplicities) of the transformation:

?

(1)

(р) $\varphi^{2}$ ;

(2)

(р) $\varphi^{m}(m$ is a natural number)

(3)

$\varphi^{ - 1}$ (assuming that $\varphi$ is invertible)

(4)

$p(\varphi )$ , where $p(t)$ is an arbitrary polynomial (assuming that $\lambda_{1}, \ldots , \lambda_{n}$ are distinct)?

Problem 24.27

Let the characteristic polynomials of the square matrices $A$ and $B$ have simple roots $\lambda_{1}, \ldots , \lambda_{m}$ and $\mu_{1}, \ldots , \mu_{n}$ respectively. Find the characteristic numbers of the Kronecker product $A\otimes B$ of the matrices $A, B$ . (See the introduction to 15.)

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Problem 24.28

(1)(2)

Let a linear transformation $\varphi$ of a vector space $\mathcal{L}$ be diagonalizable. Prove the statements:

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(1)

$\operatorname {Im} \varphi$ is the linear span of the set of all eigenvectors corresponding to nonzero eigenvalues

(2)

$\mathcal{L} = \operatorname {Im} \varphi \oplus \operatorname {Ker} \varphi$ .

Problem 24.29

Give an example of a linear transformation $\varphi$ of the space $\mathcal{R}_{n}$ for which $\mathcal{R}_{n} \neq \operatorname {Im} \varphi + \operatorname {Ker} \varphi$ .

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Problem 24.30

A linear transformation of a real $n$ -dimensional vector space is given by its matrix. Compute the eigenvalues and find a maximal linearly independent system of eigenvectors of the transformation. If the system of vectors found forms a basis, write the matrix of the transformation in it and determine the geometric meaning of the transformation (in items 1-10 $n = 2$ , in items 11-30 $n = 3$ , in items 31-40 $n = 4$ ):

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(1)

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

(2)

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

(3)

$\mathbf{A}_{36}$ . $\mathbf{A}_{36} = \left[\begin{smallmatrix} 0 & 2 \\ -1 & -3 \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{47}$ . $\mathbf{A}_{47} = \left[\begin{smallmatrix} 5 & -2 \\ -2 & 8 \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{48}$ . $\mathbf{A}_{48} = \left[\begin{smallmatrix} 4/3 & -1 \\ -1 & 3/4 \end{smallmatrix}\right]$ .

(6)

$\mathbf{A}_{13}$ . $\mathbf{A}_{13} = \left[\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right]$ .

(7)

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

(8)

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

(9)

$\mathbf{A}_{30}$ . $\mathbf{A}_{30} = \left[\begin{smallmatrix} 25 & 60 \\ 60 & 144 \end{smallmatrix}\right]$ .

(10)

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

(11)

$\mathbf{A}_{261}$ . $\mathbf{A}_{261} = \left[\begin{smallmatrix} 1 & 0 & 3 \\ 2 & 1 & 2 \\ 3 & 0 & 1 \end{smallmatrix}\right]$ .

(12)

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

(13)

$\mathbf{A}_{290}$ . $\mathbf{A}_{290} = \left[\begin{smallmatrix} 2 & 1 & -3 \\ 3 & -2 & -3 \\ 1 & 1 & -2 \end{smallmatrix}\right]$ .

(14)

$\mathbf{A}_{291}$ . $\mathbf{A}_{291} = \left[\begin{smallmatrix} 2 & -1 & 1 \\ 1 & 2 & -1 \\ 1 & -1 & 2 \end{smallmatrix}\right]$ .

(15)

$\mathbf{A}_{292}$ . $\mathbf{A}_{292} = \left[\begin{smallmatrix} 1 & -2 & -1 \\ -1 & 1 & 1 \\ 1 & 0 & -1 \end{smallmatrix}\right]$ .

(16)

$\mathbf{A}_{293}$ . $\mathbf{A}_{293} = \left[\begin{smallmatrix} 1 & 10 & 3 \\ 2 & 1 & 2 \\ 3 & 10 & 1 \end{smallmatrix}\right]$ .

(17)

$\mathbf{A}_{294}$ . $\mathbf{A}_{294} = \left[\begin{smallmatrix} 2 & -2 & 0 \\ -2 & 9 & 2 \\ 0 & 2 & 2 \end{smallmatrix}\right]$ .

(18)

$\mathbf{A}_{295}$ . $\mathbf{A}_{295} = \left[\begin{smallmatrix} 4 & 2 & -6 \\ 2 & 1 & -3 \\ -6 & -3 & 9 \end{smallmatrix}\right]$ .

(19)

$\mathbf{A}_{221}$ . $\mathbf{A}_{221} = \left[\begin{smallmatrix} 4 & 1 & -1 \\ 2 & 5 & -2 \\ 4 & 4 & -1 \end{smallmatrix}\right]$ .

(20)

$\mathbf{A}_{267}$ . $\mathbf{A}_{267} = \left[\begin{smallmatrix} -1 & 3 & -1 \\ -3 & 5 & -1 \\ -3 & 3 & 1 \end{smallmatrix}\right]$ .

(21)

$\mathbf{A}_{296}$ . $\mathbf{A}_{296} = \left[\begin{smallmatrix} 3 & -2 & 6 \\ -2 & 6 & 3 \\ 6 & 3 & -2 \end{smallmatrix}\right]$ .

(22)

$\mathbf{A}_{297}$ . $\mathbf{A}_{297} = \left[\begin{smallmatrix} 0 & 0 & 0 \\ 0 & -1 & 1 \\ 0 & 0 & -1 \end{smallmatrix}\right]$ .

(23)

$\mathbf{A}_{298}$ . $\mathbf{A}_{298} = \left[\begin{smallmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 2 & 2 & 1 \end{smallmatrix}\right]$ .

(24)

$\mathbf{A}_{273}$ . $\mathbf{A}_{273} = \left[\begin{smallmatrix} -1 & 4 & 4 \\ -10 & -18 & -20 \\ 9 & 13 & 15 \end{smallmatrix}\right]$ .

(25)

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

(26)

$\mathbf{A}_{230}$ . $\mathbf{A}_{230} = \left[\begin{smallmatrix} -1 & 4 & 3 \\ -2 & 5 & 3 \\ 2 & -4 & -2 \end{smallmatrix}\right]$ .

(27)

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

(28)

$\mathbf{A}_{299}$ . $\mathbf{A}_{299} = \left[\begin{smallmatrix} -2 & -2 & -3 \\ 2 & 3 & 6 \\ -1 & -2 & -4 \end{smallmatrix}\right]$ .

(29)

$\mathbf{A}_{287}$ . $\mathbf{A}_{287} = \left[\begin{smallmatrix} 7 & -12 & 6 \\ 10 & -19 & 10 \\ 12 & -24 & 13 \end{smallmatrix}\right]$ .

(30)

$\mathbf{A}_{283}$ . $\mathbf{A}_{283} = \left[\begin{smallmatrix} 2 & 5 & 1 \\ -1 & -3 & 0 \\ -2 & -3 & -2 \end{smallmatrix}\right]$ .

(31)

$\mathbf{A}_{477}$ . $\mathbf{A}_{477} = \left[\begin{smallmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{smallmatrix}\right]$ .

(32)

$\mathbf{A}_{479}$ . $\mathbf{A}_{479} = \left[\begin{smallmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{smallmatrix}\right]$ .

(33)

$\mathbf{A}_{480}$ . $\mathbf{A}_{480} = \left[\begin{smallmatrix} 5 & -2 & 4 & -4 \\ -2 & 8 & 2 & -2 \\ 0 & 0 & 4 & 5 \\ 0 & 0 & 5 & 4 \end{smallmatrix}\right]$ .

(34)

$\mathbf{A}_{481}$ . $\mathbf{A}_{481} = \left[\begin{smallmatrix} 1 & -1 & 1 & -1 \\ -1 & 1 & -1 & 1 \\ 1 & -1 & 1 & -1 \\ -1 & 1 & -1 & 1 \end{smallmatrix}\right]$ .

(35)

$\mathbf{A}_{482}$ . $\mathbf{A}_{482} = \left[\begin{smallmatrix} 1 & 0 & 0 & -1 \\ 0 & 1 & -1 & 0 \\ -1 & 0 & 0 & 1 \\ 0 & -1 & 1 & 0 \end{smallmatrix}\right]$ .

(36)

$\mathbf{A}_{449}$ . $\mathbf{A}_{449} = \left[\begin{smallmatrix} -1 & 2 & -4 & -2 \\ -2 & -2 & -9 & -7 \\ 4 & -9 & 0 & -5 \\ -2 & 7 & 5 & 8 \end{smallmatrix}\right]$ .

(37)

$\mathbf{A}_{483}$ . $\mathbf{A}_{483} = \left[\begin{smallmatrix} 2 & 1 & 1 & 4 \\ -2 & -1 & -2 & -2 \\ -2 & -2 & -1 & -2 \\ 1 & 1 & 1 & -1 \end{smallmatrix}\right]$ .

(38)

$\mathbf{A}_{484}$ . $\mathbf{A}_{484} = \left[\begin{smallmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & 1 & -1 \\ 1 & -1 & -1 & 1 \end{smallmatrix}\right]$ .

(39)

$\mathbf{A}_{485}$ . $\mathbf{A}_{485} = \left[\begin{smallmatrix} 3 & -1 & -1 & -1 \\ 1 & 1 & 1 & -3 \\ 1 & 1 & -3 & 1 \\ -1 & 3 & -1 & -1 \end{smallmatrix}\right]$ .

(40)

$\mathbf{A}_{469}$ . $\mathbf{A}_{469} = \left[\begin{smallmatrix} -2 & -1 & 3 & -7 \\ -3 & 0 & 3 & -7 \\ 3 & 1 & -2 & 7 \\ 3 & 1 & -3 & 8 \end{smallmatrix}\right]$ .

Problem 24.31

A linear transformation of a complex $n$ -dimensional vector space is given by its matrix. Find a basis of eigenvectors and write the matrix of the transformation in this basis:

?

(1)

$\mathbf{A}_{20}$ . $\mathbf{A}_{20} = \left[\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{50}$ . $\mathbf{A}_{50} = \left[\begin{smallmatrix} 0 & -1 \\ 1 & -1 \end{smallmatrix}\right]$ .

(3)

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

(4)

$\mathbf{A}_{79} \quad \left(\varepsilon = e^{2\pi i / 3}\right)$ . $\mathbf{A}_{79} = \left[\begin{smallmatrix} 0 & \varepsilon \\ \varepsilon^{-1} & 0 \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{94}$ . $\mathbf{A}_{94} = \left[\begin{smallmatrix} 1 & -i \\ -i & 1 \end{smallmatrix}\right]$ .

(6)

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

(7)

$\mathbf{A}_{78} \quad \left(\varepsilon = e^{2\pi i / 3}\right)$ . $\mathbf{A}_{78} = \left[\begin{smallmatrix} \varepsilon & 1 \\ -1 & \varepsilon \end{smallmatrix}\right]$ .

(8)

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

(9)

$\mathbf{A}_{239}$ . $\mathbf{A}_{239} = \left[\begin{smallmatrix} 0 & 1 & 2 \\ -1 & 0 & -2 \\ -2 & 2 & 0 \end{smallmatrix}\right]$ .

(10)

$\mathbf{A}_{262}$ . $\mathbf{A}_{262} = \left[\begin{smallmatrix} -1 & -2 & 2 \\ -2 & -1 & 2 \\ -3 & -2 & 3 \end{smallmatrix}\right]$ .

(11)

$\mathbf{A}_{263}$ . $\mathbf{A}_{263} = \left[\begin{smallmatrix} 3 & -3 & 1 \\ 3 & -2 & 2 \\ -1 & 2 & 0 \end{smallmatrix}\right]$ .

(12)

$\mathbf{A}_{300}$ . $\mathbf{A}_{300} = \left[\begin{smallmatrix} 2 & 4 & -4 \\ 0 & 5 & -3 \\ -1 & 3 & 1 \end{smallmatrix}\right]$ .

(13)

$\mathbf{A}_{301}$ . $\mathbf{A}_{301} = \left[\begin{smallmatrix} 0 & -6 & -2 \\ -2 & -4 & -2 \\ 2 & 11 & 4 \end{smallmatrix}\right]$ .

(14)

$\mathbf{A}_{260}$ . $\mathbf{A}_{260} = \left[\begin{smallmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{smallmatrix}\right]$ .

(15)

$\mathbf{A}_{363}\left(\omega = e^{2\pi i / 3}\right)$ . $\mathbf{A}_{363} = \left[\begin{smallmatrix} 1 & 1 & 1 \\ 1 & \omega & \omega^{2} \\ 1 & \omega^{2} & \omega \end{smallmatrix}\right]$ .

(16)

$\mathbf{A}_{376}$ . $\mathbf{A}_{376} = \left[\begin{smallmatrix} 1+i & -1-i & 2+2 i \\ 0 & i & 2 \\ 0 & -1 & 3+i \end{smallmatrix}\right]$ .

(17)

$\mathbf{A}_{377}$ . $\mathbf{A}_{377} = \left[\begin{smallmatrix} 1 & 1-i & 0 \\ 1+i & 3 & i \\ 0 & -i & 1 \end{smallmatrix}\right]$ .

(18)

$\mathbf{A}_{432}$ . $\mathbf{A}_{432} = \left[\begin{smallmatrix} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{smallmatrix}\right]$ .

(19)

$\mathbf{A}_{447}$ . $\mathbf{A}_{447} = \left[\begin{smallmatrix} -1 & 2 & 0 & 1 \\ 1 & -1 & 2 & 0 \\ 0 & 1 & -1 & 2 \\ 2 & 0 & 1 & -1 \end{smallmatrix}\right]$ .

(20)

$\mathbf{A}_{486}$ . $\mathbf{A}_{486} = \left[\begin{smallmatrix} i & 1 & 0 & 0 \\ 1 & i & 0 & 0 \\ 0 & 0 & i & 1 \\ 0 & 0 & 1 & i \end{smallmatrix}\right]$ .

(21)

$\mathbf{A}_{472}$ . $\mathbf{A}_{472} = \left[\begin{smallmatrix} 1 & 1 & 1 & 1 \\ 1 & i & -1 & -i \\ 1 & -1 & 1 & -1 \\ 1 & -i & -1 & i \end{smallmatrix}\right]$ .

Problem 24.32

Find the eigenvalues and a maximal linearly independent system of eigenvectors of the linear transformation given by its matrix. Explain why the transformation is not diagonalizable:

?

(1)

$\mathbf{A}_{51}$ . $\mathbf{A}_{51} = \left[\begin{smallmatrix} -7 & -2 \\ 8 & 1 \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{52}$ . $\mathbf{A}_{52} = \left[\begin{smallmatrix} 11 & -2 \\ 18 & -1 \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{286}$ . $\mathbf{A}_{286} = \left[\begin{smallmatrix} 5 & 4 & 4 \\ 2 & 6 & 4 \\ -3 & -5 & -3 \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{303}$ . $\mathbf{A}_{303} = \left[\begin{smallmatrix} 0 & 3 & 3 \\ -1 & 8 & 6 \\ 2 & -14 & -10 \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{289}$ . $\mathbf{A}_{289} = \left[\begin{smallmatrix} -2 & 1 & 2 \\ -1 & 0 & 2 \\ -2 & 0 & 3 \end{smallmatrix}\right]$ .

(6)

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

(7)

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

(8)

$\mathbf{A}_{487}$ . $\mathbf{A}_{487} = \left[\begin{smallmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 3 & 4 & 0 & 0 \\ -1 & -1 & 0 & 0 \end{smallmatrix}\right]$ .

(9)

$\mathbf{A}_{548}$ . $\mathbf{A}_{548} = \left[\begin{smallmatrix} 1 & 0 & 0 & 1 & -1 \\ 0 & 1 & -2 & 3 & -3 \\ 0 & 0 & -1 & 2 & -2 \\ 1 & -1 & 1 & 0 & 1 \\ 1 & -1 & 1 & -1 & 2 \end{smallmatrix}\right]$ .

Problem 24.33

Find the characteristic numbers of the linear transformation given by its matrix. Determine whether the transformation is diagonalizable:

(1) in the real space,

(2) in the complex space. If yes, find a basis of eigenvectors and write the matrix of the transformation in it; otherwise indicate which of the necessary conditions for diagonalizability is not satisfied:

?

(1)

$\mathbf{A}_{45}$ . $\mathbf{A}_{45} = \left[\begin{smallmatrix} 1 & -1 \\ 3 & -2 \end{smallmatrix}\right]$ .

(2)

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

(3)

$\mathbf{A}_{259}$ . $\mathbf{A}_{259} = \left[\begin{smallmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{44}$ . $\mathbf{A}_{44} = \left[\begin{smallmatrix} 5 & -1 \\ 4 & 1 \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{253}$ . $\mathbf{A}_{253} = \left[\begin{smallmatrix} 0 & 1 & 1 \\ -1 & 0 & -1 \\ -1 & 1 & 0 \end{smallmatrix}\right]$ .

(6)

$\mathbf{A}_{98}$ . $\mathbf{A}_{98} = \left[\begin{smallmatrix} 0 & 1 \\ 1 & 2 i \end{smallmatrix}\right]$ .

(7)

$\mathbf{A}_{436}$ . $\mathbf{A}_{436} = \left[\begin{smallmatrix} 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{smallmatrix}\right]$ .

(8)

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

(9)

$\mathbf{A}_{478}$ . $\mathbf{A}_{478} = \left[\begin{smallmatrix} 1 & 0 & 0 & -1 \\ 0 & 1 & -1 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 \end{smallmatrix}\right]$ .

(10)

$\mathbf{A}_{474}$ . $\mathbf{A}_{474} = \left[\begin{smallmatrix} -5 & -2 & -3 & -1 \\ 4 & 2 & 2 & 1 \\ 6 & 3 & 3 & 1 \\ 2 & -3 & 4 & 0 \end{smallmatrix}\right]$ .

Problem 24.34

Solve the eigenvalue-eigenvector problem and indicate the diagonal form of the matrix of the linear transformation given in the standard basis (real $n$ -dimensional arithmetic space in items 1-10, complex $n$ -dimensional arithmetic space in items 11-13):

?

(1)

$\mathbf{A}_{604}$ . $\mathbf{A}_{604} = \left[\begin{smallmatrix} 0 & 0 & \cdots & 0 & 0 & 1 \\ 0 & 0 & \cdots & 0 & 1 & 0 \\ 0 & 0 & \cdots & 1 & 0 & 0 \\ \vdots & \vdots & \iddots & \vdots & \vdots & \vdots \\ 0 & 1 & \cdots & 0 & 0 & 0 \\ 1 & 0 & \cdots & 0 & 0 & 0 \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{621} \quad (n = 2m)$ . $\mathbf{A}_{621} = \left[\begin{smallmatrix} 1 & 0 & 0 & \ldots & 0 & 0 & 2 \\ 0 & 1 & 0 & \ldots & 0 & 2 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 2 & 0 & \ldots & 0 & 1 & 0 \\ 2 & 0 & 0 & \ldots & 0 & 0 & 1 \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{625}$ . $\mathbf{A}_{625} = \left[\begin{smallmatrix} 1 & 2 & 2 & \ldots & 2 & 2 \\ 2 & 1 & 2 & \ldots & 2 & 2 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 2 & 2 & 2 & \ldots & 1 & 2 \\ 2 & 2 & 2 & \ldots & 2 & 1 \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{627}$ . $\mathbf{A}_{627} = \left[\begin{smallmatrix} x & y & y & \ldots & y \\ y & x & y & \ldots & y \\ \ldots & \ldots & \ldots & \ldots & . \\ y & y & y & \ldots & x \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{640}$ . $\mathbf{A}_{640} = \left[\begin{smallmatrix} n-1 & 1 & 0 & 0 & \ldots & 0 & 0 & 0 \\ -n+1 & n-3 & 2 & 0 & \ldots & 0 & 0 & 0 \\ 0 & -n+2 & n-5 & 3 & \ldots & 0 & 0 & 0 \\ \ldots \ldots \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \ldots \ldots \ldots \ldots \\ 0 & 0 & 0 & 0 & \ldots & -2 & -n+3 & n-1 \\ 0 & 0 & 0 & 0 & \ldots & 0 & -1 & -n+1 \end{smallmatrix}\right]$ .

(6)

$\mathbf{A}_{639}$ . $\mathbf{A}_{639} = \left[\begin{smallmatrix} 0 & 1 & 0 & 0 & \ldots & 0 & 0 \\ n-1 & 0 & 2 & 0 & \ldots & 0 & 0 \\ 0 & n-2 & 0 & 3 & \ldots & 0 & 0 \\ \ldots \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \ldots \\ 0 & 0 & 0 & 0 & \ldots & 0 & n-1 \\ 0 & 0 & 0 & 0 & \ldots & 1 & 0 \end{smallmatrix}\right]$ .

(7)

$\mathbf{A}_{634}$ . $\mathbf{A}_{634} = \left[\begin{smallmatrix} 0 & 1 & 0 & 0 & \ldots & 0 & 0 \\ 1 & 0 & 1 & 0 & \ldots & 0 & 0 \\ 0 & 1 & 0 & 1 & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & 0 & \ldots & 0 & 1 \\ 0 & 0 & 0 & 0 & \ldots & 1 & 0 \end{smallmatrix}\right]$ .

(8)

$\mathbf{A}_{620}$ . $\mathbf{A}_{620} = \left[\begin{smallmatrix} 1 & -1 & \ldots & (-1)^{n-1} \\ -1 & 1 & \ldots & (-1)^{n-2} \\ \ldots \ldots \ldots \ldots \ldots \ldots \ldots & \ldots & \ldots \ldots \\ (-1)^{n-1} & (-1)^{n-2} & \ldots & 1 \end{smallmatrix}\right]$ .

(9)

$\mathbf{A}_{606}(\lambda_{1} = \ldots = \lambda_{m} = 1, \lambda_{m + 1} = \ldots = \lambda_{n} = 2; \quad m = \left[(n + 1) / 2\right])$ . $\mathbf{A}_{606} = \left[\begin{smallmatrix} 0 & 0 & \cdots & 0 & 0 & \lambda^2_1 \\ 0 & 0 & \cdots & 0 & \lambda^2_2 & 0 \\ 0 & 0 & \cdots & \lambda^2_3 & 0 & 0 \\ \vdots & \vdots & \iddots & \vdots & \vdots & \vdots \\ 0 & \lambda^2_{n-1} & \cdots & 0 & 0 & 0 \\ \lambda^2_n & 0 & \cdots & 0 & 0 & 0 \end{smallmatrix}\right]$ .

(10)

$\mathbf{A}_{606}(\lambda_{1} = \ldots = \lambda_{m} = 1, \lambda_{m + 1} = \ldots = \lambda_{n} = 0; \quad m = \left[(n + 1) / 2\right])$ . $\mathbf{A}_{606} = \left[\begin{smallmatrix} 0 & 0 & \cdots & 0 & 0 & \lambda^2_1 \\ 0 & 0 & \cdots & 0 & \lambda^2_2 & 0 \\ 0 & 0 & \cdots & \lambda^2_3 & 0 & 0 \\ \vdots & \vdots & \iddots & \vdots & \vdots & \vdots \\ 0 & \lambda^2_{n-1} & \cdots & 0 & 0 & 0 \\ \lambda^2_n & 0 & \cdots & 0 & 0 & 0 \end{smallmatrix}\right]$ .

(11)

$\mathbf{A}_{605}(\lambda_{1} = \ldots = \lambda_{m} = 1, \lambda_{m + 1} = \ldots = \lambda_{n} = - 1; m = \left[(n + 1) / 2\right])$ . $\mathbf{A}_{605} = \left[\begin{smallmatrix} 0 & 0 & \cdots & 0 & 0 & \lambda_1 \\ 0 & 0 & \cdots & 0 & \lambda_2 & 0 \\ 0 & 0 & \cdots & \lambda_3 & 0 & 0 \\ \vdots & \vdots & \iddots & \vdots & \vdots & \vdots \\ 0 & \lambda_{n-1} & \cdots & 0 & 0 & 0 \\ \lambda_n & 0 & \cdots & 0 & 0 & 0 \end{smallmatrix}\right]$ .

(12)

$\mathbf{A}_{614}$ . $\mathbf{A}_{614} = \left[\begin{smallmatrix} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & . \\ 0 & 0 & 0 & \ldots & 1 \\ 1 & 0 & 0 & \ldots & 0 \end{smallmatrix}\right]$ .

(13)

$\mathbf{A}_{635}$ . $\mathbf{A}_{635} = \left[\begin{smallmatrix} 0 & 1 & 0 & 0 & \ldots & 0 & 0 \\ -1 & 0 & 1 & 0 & \ldots & 0 & 0 \\ 0 & -1 & 0 & 1 & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & 0 & \ldots & 0 & 1 \\ 0 & 0 & 0 & 0 & \ldots & -1 & 0 \end{smallmatrix}\right]$ .

Problem 24.35

Find the characteristic numbers of the matrix:

?

(1)

$\mathbf{A}_{490}$ . $\mathbf{A}_{490} = \left[\begin{smallmatrix} 0 & 1 & 0 & 0 \\ -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \\ 0 & 0 & -1 & 0 \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{491}$ . $\mathbf{A}_{491} = \left[\begin{smallmatrix} 3 & 1 & 0 & 2 i \\ 1 & 3 & -2 i & 0 \\ 0 & 2 i & 1 & 1 \\ -2 i & 0 & 1 & 1 \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{492}$ . $\mathbf{A}_{492} = \left[\begin{smallmatrix} 7 & 3 & 1+2 i & -1+2 i \\ 3 & 7 & 1-2 i & -1-2 i \\ 1-2 i & 1+2 i & 7 & -3 \\ -1-2 i & -1+2 i & -3 & 7 \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{549}$ . $\mathbf{A}_{549} = \left[\begin{smallmatrix} 0 & 2 & -2 & 0 & -4 \\ -2 & 0 & 4 & -5 & 3 \\ 2 & -4 & 0 & 1 & 1 \\ 0 & 5 & -1 & 0 & 2 \\ 4 & -3 & -1 & -2 & 0 \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{550}$ . $\mathbf{A}_{550} = \left[\begin{smallmatrix} 0 & 0 & -1 & 1 & 0 \\ 1 & 0 & -1 & 1 & 0 \\ 1 & 0 & -1 & 0 & 1 \\ 0 & 1 & -1 & 0 & 1 \\ 0 & 1 & -1 & 0 & 0 \end{smallmatrix}\right]$ .

(6)

$\mathbf{A}_{638}$ . $\mathbf{A}_{638} = \left[\begin{smallmatrix} a & b & 0 & \ldots & 0 & 0 & 0 \\ b & a & b & \ldots & 0 & 0 & 0 \\ 0 & b & a & \ldots & 0 & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \cdots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & b & a & b \\ 0 & 0 & 0 & \ldots & 0 & b & a \end{smallmatrix}\right]$ .

(7)

$\mathbf{A}_{643}$ . $\mathbf{A}_{643} = \left[\begin{smallmatrix} a_1 & a_2 & \ldots & a_n \\ a_n & a_1 & \ldots & a_{n-1} \\ \ldots \ldots & \ldots & \ldots & \ldots \\ R a_2 & a_3 & \ldots & a_1 \end{smallmatrix}\right]$ .

(8)

$\mathbf{A}_{642}$ . $\mathbf{A}_{642} = \left[\begin{smallmatrix} a_{1} & a_{2} & \ldots & a_{n-1} & a_{n} \\ a_{2} & a_{3} & \ldots & a_{n} & a_{1} \\ \ldots & \ldots & \ldots & \ldots \ldots & \ldots \ldots \\ a_{n} & a_{1} & \ldots & a_{n-2} & a_{n-1} \end{smallmatrix}\right]$ .

(9)

$\mathbf{A}_{645}(n$ is odd). $\mathbf{A}_{645} = \left[\begin{smallmatrix} 1 & 1 & 1 & \ldots & 1 \\ 1 & \varepsilon & \varepsilon^2 & \ldots & \varepsilon^{n-1} \\ 1 & \varepsilon^2 & \varepsilon^4 & \ldots & \varepsilon^{2(n-1)} \\ \ldots \ldots \ldots \ldots & \ldots \ldots & \ldots \ldots & \ldots \ldots \\ 1 & \varepsilon^{n-1} & \varepsilon^{2(n-1)} & \ldots & \varepsilon^{(n-1)^2} \end{smallmatrix}\right]$ .

Problem 24.36

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

Compute:

?

(1)

$2^{n + 1} \prod_{k = 1}^{n} \cos \frac{\pi k}{n + 1}$

(2)

$\sum_{k = 1}^{n} \cos \frac{\pi k}{n + 1}$

(3)

$\sum_{k = 0}^{n - 1} \varepsilon^{k^{2}}$ , where $\varepsilon = e^{2\pi i / n}, \quad n = 2m + 1$

(4)

$\prod_{0 \leqslant j < k \leqslant n - 1}\left(\varepsilon^{k} - \varepsilon^{j}\right)$ , where $\varepsilon = e^{2\pi i / n}, \quad n = 2m + 1$ .

Problem 24.37

(1)(2)(3)

?

(1)

One of the matrices $\mathbf{A}_{287}, \mathbf{A}_{289}$ is similar to the matrix $D = \operatorname {diag}(1, 1, - 1)$ . Which one? Justify the answer without finding the eigenvectors and characteristic numbers. $\mathbf{A}_{287} = \left[\begin{smallmatrix} 7 & -12 & 6 \\ 10 & -19 & 10 \\ 12 & -24 & 13 \end{smallmatrix}\right], \mathbf{A}_{289} = \left[\begin{smallmatrix} -2 & 1 & 2 \\ -1 & 0 & 2 \\ -2 & 0 & 3 \end{smallmatrix}\right]$ .

(2)

The matrix $D = \operatorname {diag}(1, 1, 0)$ is similar to one of the matrices $\mathbf{A}_{230}, \mathbf{A}_{264}$ . Determine which one, without finding the eigenvalues and eigenvectors. $\mathbf{A}_{230} = \left[\begin{smallmatrix} -1 & 4 & 3 \\ -2 & 5 & 3 \\ 2 & -4 & -2 \end{smallmatrix}\right], \mathbf{A}_{264} = \left[\begin{smallmatrix} 0 & 1 & 1 \\ 1 & 1 & 0 \\ -1 & 0 & 1 \end{smallmatrix}\right]$ .

(3)

Of the two matrices $\mathbf{A}_{304}, \mathbf{A}_{305}$ , one is similar to the matrix $D_{1} = \operatorname {diag}(1, - 1, 0)$ , and the other to the matrix $D_{2} = \operatorname {diag}(1, 1, 0)$ . Determine which is which, without computing the eigenvalues and eigenvectors. $\mathbf{A}_{304} = \left[\begin{smallmatrix} 2 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & 1 & 0 \end{smallmatrix}\right], \mathbf{A}_{305} = \left[\begin{smallmatrix} 2 & -1 & -1 \\ 1 & 0 & -1 \\ 3 & -1 & -2 \end{smallmatrix}\right]$ .

Problem 24.38

(1)(2)

?

(1)

The matrix $\mathbf{A}_{283}$ is similar to one of the matrices: $- E$ , $J_{3}(- 1), \operatorname {diag}\left(- 1, J_{2}(- 1)\right)$ . Which one? $\mathbf{A}_{283} = \left[\begin{smallmatrix} 2 & 5 & 1 \\ -1 & -3 & 0 \\ -2 & -3 & -2 \end{smallmatrix}\right]$ .

(2)

One of the matrices $\mathbf{A}_{457}, \mathbf{A}_{458}, \mathbf{A}_{485}$ is similar to the matrix $J_{4}(0)$ . Which one? $\mathbf{A}_{457} = \left[\begin{smallmatrix} 1 & 0 & 0 & -1 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & -1 & 0 \\ 0 & 1 & -1 & 0 \end{smallmatrix}\right], \mathbf{A}_{458} = \left[\begin{smallmatrix} 1 & 0 & 0 & -1 \\ 0 & 1 & -1 & 0 \\ 0 & 1 & -1 & 0 \\ 1 & 0 & 0 & -1 \end{smallmatrix}\right], \mathbf{A}_{485} = \left[\begin{smallmatrix} 3 & -1 & -1 & -1 \\ 1 & 1 & 1 & -3 \\ 1 & 1 & -3 & 1 \\ -1 & 3 & -1 & -1 \end{smallmatrix}\right]$ .

Problem 24.39

The triangles $ABC$ and $A^{\prime } B^{\prime } C^{\prime }$ are similar (with similarity coefficient $\lambda$ ). If the side lengths of the triangle $ABC$ equal $a, b, c$ , then the corresponding sides of the triangle $A^{\prime } B^{\prime } C^{\prime }$ have lengths $3a + b + c, a + 3b + c, a + b + 3c$ . Prove that the triangles are equilateral, and find $\lambda$ .

?

Problem 24.40

The sum of distinct natural numbers $n_{1}, n_{2}, n_{3}, n_{4}$ equals 18. After they were increased by the same factor $\lambda$ , the numbers $n_{1} + n_{2} + n_{3} + n_{4}, n_{1} + n_{2} - n_{3} - n_{4}$ , $n_{1} - n_{2} + n_{3} - n_{4}, \quad n_{1} - 2n_{2} - n_{3} + 3n_{4}$ were obtained. Find $n_{1}, n_{2}, n_{3}, n_{4}, \lambda$ .

?

Problem 24.41

The sequence $\left\{ x_{n}\right\}$ is given by the recurrence formula: $x_{n + 1} = \frac{2}{3} x_{n} + \frac{1}{3} x_{n - 1} \quad (n = 1, 2, \ldots ); x_{0} = a, \quad x_{1} = b$ . Prove that the sequence converges, and find its limit.

?

Problem 24.42

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

Find the eigenvalues and eigenvectors (eigenfunctions) of differentiation $D$ as a linear transformation of each of the following vector spaces of real functions ( $n$ a fixed natural number):

?

(1)

the space of all polynomials of degree at most $n$

(2)

the space of all trigonometric polynomials of the form $f(t) = a_{0} + a_{1} \cos t + b_{1} \sin t + \ldots + a_{n} \cos nt + b_{n} \sin nt$

(3)

the linear span of the system of functions $e^{\lambda_{1} t}, \ldots , e^{\lambda_{n} t}$ , where $\lambda_{1}, \ldots , \lambda_{n}$ are pairwise distinct numbers.

(4)

the space of all functions of the form $e^{\lambda_{0} t} p(t)$ , where $p(t)$ is any polynomial of degree at most $n$ , $\lambda_{0}$ a fixed number $\left(\lambda_{0} \neq 0\right)$ .

Problem 24.43

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

Find the eigenvalues and eigenvectors of the transformation $D^{2}$ in the spaces of functions from problem 24.42:

?

(1)

the space of all polynomials of degree at most $n$

(2)

the space of all trigonometric polynomials of the form $f(t) = a_{0} + a_{1} \cos t + b_{1} \sin t + \ldots + a_{n} \cos nt + b_{n} \sin nt$

(3)

the linear span of the system of functions $e^{\lambda_{1} t}, \ldots , e^{\lambda_{n} t}$ , where $\lambda_{1}, \ldots , \lambda_{n}$ are pairwise distinct numbers.

(4)

the space of all functions of the form $e^{\lambda_{0} t} p(t)$ , where $p(t)$ is any polynomial of degree at most $n$ , $\lambda_{0}$ a fixed number $\left(\lambda_{0} \neq 0\right)$ .

Problem 24.44

(1)(2)(3)

Verify that functions of the form $y = e^{t} p(t)$ , where $p(t)$ is a polynomial of degree at most two, form a vector space $\mathcal{L}$ . Verify that $\varphi$ is a linear transformation of the space $\mathcal{L}$ , and solve the eigenvalue-eigenvector problem for $\varphi$ :

?

(1)

$\varphi (y) = y^{\prime \prime } - 2y^{\prime } + y$ , i.e. $\varphi = D^{2} - 2D + \iota$

(2)

$\varphi = D^{3} - 2D^{2}$

(3)

$\varphi = D^{3} - 3D^{2} + 3D + \iota$ .

Problem 24.45

(1)(2)

Verify that functions of the form $y = e^{ - t}(a\cos t + b\sin t)$ form a vector space $\mathcal{M}$ , and that the transformation $\varphi = p(D)$ , where $p(t)$ is a given polynomial and $D$ is differentiation, is a linear transformation of the space $\mathcal{M}$ . Solve the eigenvalue-eigenvector problem for $\varphi$ , if:

?

(1)

$p(t) = (t + 1)^{2}$

(2)

$p(t) = t^{2} - 1$

Problem 24.46

(1)(2)

In the linear span of the functions $\cos 2t, \sin 2t, t\cos 2t$ , $t\sin 2t$ , a linear transformation $\varphi = p(D)$ is given, where $p(t)$ is a polynomial and $D$ is differentiation. Solve the eigenvalue-eigenvector problem for $\varphi$ , if:

?

(1)

$p(t) = t^{2} + 4$

(2)

$p(t) = t^{4} + 8t^{2}$ .

Problem 24.47

(1)(2)(3)

Find the eigenvalues and eigenvectors of the linear transformation $\varphi$ of the space of real polynomials $p(t)$ of degree at most two, if:

?

(1)

$\varphi (p) = tp^{\prime }$ ; 1

(2)

$\varphi (p) = (tp)^{\prime }$

(3)

$\varphi (p) = t^{2} p^{\prime \prime } - tp^{\prime } + 2p$ .

Problem 24.48

Find the eigenvalues and eigenvectors of the differentiation transformation in the space of functions infinitely differentiable on the whole real line.

?

Problem 24.49

(1)(2)

Let $\mathcal{L}$ be the set of functions $y(t)$ infinitely differentiable on the segment $\left[0, \pi \right]$ and such that $y(0) = y(\pi ) = 0$ .

?

(1)

Verify that $\mathcal{L}$ is a vector space.

(2)

Find the eigenvectors and eigenvalues of the linear transformation $\varphi$ of the space $\mathcal{L}$ given by the formula $\varphi (y) = y^{\prime \prime }$ .

Problem 24.50

Let $A, B$ be square matrices, and let the matrix $\left[\begin{smallmatrix} A & C \\ O & B\end{smallmatrix}\right]^{\square }$ be diagonalizable. Prove that the matrices $A, B$ are diagonalizable. Show by an example that the converse statement is not true.

?

Problem 24.51

(1)(2)(3)

Fix a real polynomial $p_{0}(t)$ of degree $m$ $(m \geqslant 1)$ . Any polynomial $p(t)$ can be divided by $p_{0}(t)$ with remainder, i.e. uniquely represented in the form

p(t) = q(t) p_{0}(t) + r(t)

(the degree of the remainder $r(t)$ is less than the degree of $p_{0}(t)$ ). Define the transformation $\varphi$ of the space $\mathcal{P}$ of all real polynomials by the formula $\varphi (p(t)) = r(t)$ .

?

(1)

Show that the transformation $\varphi$ is linear and $\varphi^{2} = \varphi$ .

(2)

Find the eigenvalues and eigenvectors of the transformation $\varphi$ .

(3)

Prove that formula (4) gives a decomposition of the space $\mathcal{P}$ into a direct sum of eigenspaces.

Problem 24.52

(1)(2)(3)

Let $\varphi$ be the operation of taking the remainder of division by the polynomial $p_{0}(t)$ (see problem 24.51) in the space of polynomials of degree at most 3. Find a basis of eigenvectors and write the matrix of the transformation $\varphi$ in this basis, if:

?

(1)

$p_{0}(t) = t$

(2)

$p_{0}(t) = t^{2} + 1$

(3)

$p_{0}(t) = (t - 1)^{3}$ .

Problem 24.53

In the space $\mathcal{R}_{n \times n}$ of square matrices of order $n$ , consider the operation of transposition $\tau : A \rightarrow A^{T}$ . Verify that $\tau$ is a linear transformation and $\tau^{2} = \iota$ . Find the eigenvalues and eigenvectors of the transformation $\tau$ . Decompose the space $\mathcal{R}_{n \times n}$ into a direct sum of eigenspaces of the transformation $\tau$ .

?

Problem 24.54

(1)(2)(3)

The set of complex matrices of order $n$ is considered as a real vector space $\mathcal{M}$ of dimension $2n^{2}$ .

?

(1)

Verify that the operation $\eta : A \rightarrow A^{H} = \bar{A}^{T}$ of Hermitian conjugation of a matrix is a linear transformation of the space $\mathcal{M}$ , and moreover $\eta^{2} = \iota$ .

(2)

Find the eigenvalues and eigenvectors of the transformation $\eta$ .

(3)

Show that the transformation $\eta$ is not a linear transformation of the complex space $\mathcal{C}_{n \times n}$ .

Problem 24.55

(1)(2)(3)

Let $A$ be a matrix of second order. The formula $\varphi (X) = AX$ defines a linear transformation of the space of matrices of second order (problem 23.47). Find the eigenvalues and a maximal linearly independent system of eigenvectors of the transformation $\varphi$ . In the case when this system is a basis, write the matrix of the transformation $\varphi$ in it:

?

(1)

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

(2)

$A = \mathbf{A}_{52}$ . $\mathbf{A}_{52} = \left[\begin{smallmatrix} 11 & -2 \\ 18 & -1 \end{smallmatrix}\right]$ .

(3)

$A = \mathbf{A}_{50}$ (in the space of complex matrices). $\mathbf{A}_{50} = \left[\begin{smallmatrix} 0 & -1 \\ 1 & -1 \end{smallmatrix}\right]$ .

Problem 24.56

(1)(2)(3)

Solve the eigenvalue-eigenvector problem for the transformation $\varphi (X) = XB$ of the space of matrices of second order, if:

?

(1)

$B = \mathbf{A}_{36}$ . $\mathbf{A}_{36} = \left[\begin{smallmatrix} 0 & 2 \\ -1 & -3 \end{smallmatrix}\right]$ .

(2)

$B = \mathbf{A}_{51}$ (the space is real). $\mathbf{A}_{51} = \left[\begin{smallmatrix} -7 & -2 \\ 8 & 1 \end{smallmatrix}\right]$ .

(3)

$B = \mathbf{A}_{45}$ (the space is complex). $\mathbf{A}_{45} = \left[\begin{smallmatrix} 1 & -1 \\ 3 & -2 \end{smallmatrix}\right]$ .

Problem 24.57

(1)(2)

A transformation of the space of matrices of second order is defined by the formula $\varphi (X) = AX - XA$ , where $A$ is a fixed matrix.

?

(1)

Show that the transformation $\varphi$ is linear, and find its matrix in the standard basis.

(2)

Solve the eigenvalue-eigenvector problem for the transformation $\varphi$ , if:

(1) $A = \mathbf{A}_{106}$

(2) $A = \mathbf{A}_{22}$ (the space is real);

(3) $A = \mathbf{A}_{20}$ (the space is complex). $\mathbf{A}_{106} = \left[\begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix}\right], \mathbf{A}_{22} = \left[\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\right], \mathbf{A}_{20} = \left[\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right]$ .

Problem 24.58

(1)(2)

Let $A$ be a nonsingular matrix of second order. Show that the formula $\varphi (X) = A^{ - 1} XA$ defines a linear transformation of the space of matrices of second order. Solve the eigenvalue-eigenvector problem for the transformation $\varphi$ , if:

?

(1)

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

(2)

$A = \mathbf{A}_{13}$ . $\mathbf{A}_{13} = \left[\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right]$ .

Problem 24.59

(1)(2)

Find the eigenvectors and eigenvalues of the shift transformation in the space of polynomials in two variables, defined in problem 23.50, if:

?

(1)

$a = 1, \quad b = 0$

(2)

$a = 1, \quad b = - 2$ .

Problem 24.60

(1)(2)(3)

Solve the problem on eigenvalues and eigenvectors for the linear transformations of the space of homogeneous polynomials of degree $n$ in two variables, defined in problem 23.51:

?

(1)

$\varphi (p) = x\frac{\partial p}{\partial y}$ ;

(2)

$\varphi (p) = y\frac{\partial p}{\partial x}$ ;

(3)

$\varphi (p) = x\frac{\partial p}{\partial x} - y\frac{\partial p}{\partial y}$ .

Problem 24.61

(1)(2)(3)

Let $A$ be a matrix of second order, $\left(x^{ * }, y^{ * }\right) = (x, y) A$ . Define the transformation $\varphi$ of the space of polynomials $p(x, y)$ of degree at most $n$ by the formula $\varphi (p(x, y)) = p\left(x^{ * }, y^{ * }\right)$ . Show that $\varphi$ is a linear transformation. Find its eigenvectors and eigenvalues, if $n = 2$ and

?

(1)

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

(2)

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

(3)

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

Problem 24.62

(1)(2)

Let $\mathcal{K}(x, y) = g_{0}(y) + xg_{1}(y) + x^{2} g_{2}(y)$ , where $g_{0}(y)$ , $g_{1}(y), g_{2}(y)$ are continuous functions on the segment $\left[ - 1, 1\right]$ . Show that the transformation $\varphi$ , defined by the formula

\varphi (p(x)) = \int _{ - 1}^{1} \mathcal{K}(x, y) p(y) dy

is a linear transformation of the space of polynomials $p(x)$ of degree at most 2. Find the eigenvalues and eigenvectors of the transformation $\varphi$ , if:

?

(1)

$\mathcal{K}(x, y) = 3x^{2} y + 5xy^{2}$

(2)

$\mathcal{K}(x, y) = y^{2} + 2x(y - 1) + \left(1 - 3y^{2}\right) x^{2}$ .

Problem 24.63

(1)(2)

Show that the formula

\varphi (f(x)) = \int _{0}^{\pi } \mathcal{K}(x, y) f(y) dy

defines a linear transformation $\varphi$ of the space of trigonometric polynomials of the form:

?

(1)

$a\cos x + b\sin x$ , if $\mathcal{K}(x, y) = \sin (x + y)$

(2)

$a + b\cos 2x + c\sin 2x$ , if $\mathcal{K}(x, y) = \cos^{2}(x - y)$ . Find the eigenvalues and eigenvectors of the transformation $\varphi$ .

Problem 24.64

Find the eigenvalues and eigenvectors of the Laplace operator $\triangle = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}$ in the space of polynomials $p(x, y)$ with real coefficients.

?

Problem 24.65

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

Let $\varphi , \psi$ be similar transformations of the vector space $\mathcal{L}$ (see formula (5) of the introduction to 23). Prove that:

?

(1)

the characteristic polynomials of the transformations $\varphi$ and $\psi$ coincide

(2)

if $x$ is an eigenvector of the transformation $\varphi$ , then $\omega^{ - 1}(x)$ is an eigenvector of the transformation $\psi$ corresponding to the same eigenvalue

(3)

if there exists in $\mathcal{L}$ a basis in which the matrix of the transformation $\varphi$ is diagonal (triangular), then an analogous basis exists also for $\psi$ .

(4)

Show by an example that the coincidence of the characteristic polynomials of two linear transformations does not imply the similarity of these transformations.