All textbooks/EXERCISES IN ALGEBRA: A Collection of Exercises in Algebra, Linear Algebra and Geometry/II · Linear Algebra and Geometry

Chapter IX

Linear operators

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39 · Definition of a linear operator. The image, the kernel, the matrix of a linear operator

40 · Eigenvectors, invariant subspaces, root subspaces

41 · Jordan canonical form and its applications. Minimal polynomial

42 · Normed vector spaces and algebras. Non-negative matrices

Section 39Definition of a linear operator. The image, the kernel, the matrix of a linear operator[24/58%]

Problem 39.1

(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)

Which of the following maps in the corresponding vector spaces are linear operators:

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(a)

$x \mapsto a$ ( $a$ is a fixed vector);

(b)

$x \mapsto x+a$ ( $a$ is a fixed vector);

(c)

$x \mapsto \alpha x$ ( $\alpha$ is a fixed scalar);

(d)

$x \mapsto (x, a) b$ ( $V$ is a Euclidean space, $a, b$ are fixed vectors);

(e)

$x \mapsto (a, x) x$ ( $V$ is a Euclidean space, $a$ is a fixed vector);

(f)

$f(x) \mapsto f(a x+b) \quad \left(f \in \mathbb {R}\left[x\right]_{n} ; a, b\right.$ are fixed numbers);

(g)

$f(x) \mapsto f(x+1)-f(x)\left(f \in \mathbb {R}\left[x\right]_{n}\right)$ ;

(h)

$f(x) \mapsto f^{(k)}(x) \quad \left(f \in \mathbb {R}\left[x\right]_{n}\right)$ ;

(i)

$\left(x_{1}, x_{2}, x_{3}\right) \mapsto \left(x_{1}+2, x_{2}+5, x_{3}\right)$ ;

(j)

$\left(x_{1}, x_{2}, x_{3}\right) \mapsto \left(x_{1}+3 x_{3}, x_{2}^{3}, x_{1}+x_{3}\right)$ ;

(k)

$\left(x_{1}, x_{2}, x_{3}\right) \mapsto \left(x_{1}, x_{2}, x_{1}+x_{2}+x_{3}\right)$ ?

Problem 39.2

Prove that every linear operator maps any linearly dependent system of vectors to a linearly dependent system.

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Problem 39.3

Prove that in an $n$ -dimensional space, for any linearly independent system of vectors $a_{1}, \ldots , a_{n}$ and an arbitrary system of vectors $b_{1}, \ldots , b_{n}$ , there exists a unique linear operator taking $a_{i}$ to $b_{i}(i=1, \ldots , n)$ .

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Problem 39.4

Prove that in a one-dimensional vector space every linear operator has the form $x \mapsto \alpha x$ , where $\alpha$ is some scalar.

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Problem 39.5

(a)(b)(c)(d)(e)(f)(g)(h)(i)

Find the images and kernels of the linear operators from problem 39.1 (for the parameter values at which the corresponding map is a linear operator):

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(a)

$x \mapsto a$ ( $a$ is a fixed vector, $a=0$ );

(b)

$x \mapsto x+a$ ( $a$ is a fixed vector, $a=0$ );

(c)

$x \mapsto \alpha x$ ( $\alpha$ is a fixed scalar);

(d)

$x \mapsto (x, a) b$ ( $V$ is a Euclidean space, $a, b$ are fixed vectors);

(e)

$x \mapsto (a, x) x$ ( $V$ is a Euclidean space, $a$ is a fixed vector, $a=0$ );

(f)

$f(x) \mapsto f(a x+b) \quad \left(f \in \mathbb {R}\left[x\right]_{n} ; a, b\right.$ are fixed numbers);

(g)

$f(x) \mapsto f(x+1)-f(x)\left(f \in \mathbb {R}\left[x\right]_{n}\right)$ ;

(h)

$f(x) \mapsto f^{(k)}(x) \quad \left(f \in \mathbb {R}\left[x\right]_{n}\right)$ ;

(i)

$\left(x_{1}, x_{2}, x_{3}\right) \mapsto \left(x_{1}, x_{2}, x_{1}+x_{2}+x_{3}\right)$ .

Problem 39.6

(a)(b)

Prove that the differentiation operator:

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(a)

is singular on the space of polynomials of degree $\leqslant n$ ;

(b)

is nonsingular on the space of functions with basis $(\cos t, \sin t)$ .

Problem 39.7

(a)(b)

Prove that every subspace of a vector space is:

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(a)

the kernel of some linear operator;

(b)

the image of some linear operator.

Problem 39.8

Prove that if two linear operators of rank 1 have equal kernels and equal images, then they commute.

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Problem 39.9

Let $\mathcal{A}-F$ -linear operator on a subspace $L$ of a space $V$ , with $L$ different from $V$ . Prove that there are infinitely many linear operators on $V$ whose restriction to the subspace $L$ coincides with $\mathcal{A}$ , provided the field $F$ is infinite.

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Problem 39.10

(a)(b)

Let $\mathcal{A}$ be a linear operator on a space $V, L$ a subspace of $V$ . Prove that:

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(a)

the image $\mathcal{A}(L)$ and the full preimage $\mathcal{A}^{-1}(L)$ are subspaces of $V$ ;

(b)

if the operator $\mathcal{A}$ is nonsingular and $V$ is finite-dimensional, then

\operatorname {dim} \mathcal{A}(L)=\operatorname {dim} \mathcal{A}^{-1}(L)=\operatorname {dim} L

Problem 39.11

Let $\mathcal{A}$ be a linear operator on a space $V, L$ a subspace of $V$ , and $L \cap \operatorname {Ker} \mathcal{A}=0$ . Prove that any linearly independent system of vectors from $L$ is taken by the operator $\mathcal{A}$ to a linearly independent system.

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Problem 39.12

Prove Frobenius's inequality for linear operators $\mathcal{A}, \mathcal{B}, \mathcal{C}$ :

\operatorname {rk}(\mathcal{B} \mathcal{A})+\operatorname {rk}(\mathcal{A} \mathcal{C}) \leqslant \operatorname {rk} \mathcal{A}+\operatorname {rk}(\mathcal{B} \mathcal{A} \mathcal{C})

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Problem 39.13

A linear operator $\mathcal{A}$ is called a pseudoreflection if $\operatorname {rk}(\mathcal{A}-\mathcal{E})=1$ .

Prove that in an $n$ -dimensional space every linear operator is a product of at most $n$ pseudoreflections.

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Problem 39.14

Prove that for a linear operator $\mathcal{A}$ on an $n$ -dimensional space the set of operators $\mathcal{X}$ such that $\mathcal{A X}=0$ is a vector space, and find its dimension.

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Problem 39.15

(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)

Find the matrix of the operator:

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(a)

$\left(x_{1}, x_{2}, x_{3}\right) \mapsto \left(x_{1}, x_{1}+2 x_{2}, x_{2}+3 x_{3}\right)$ on the space $\mathbb {R}^{3}$ in the basis of unit vectors;

(b)

of the rotation of the plane through an angle $\alpha$ in an arbitrary orthonormal basis;

(c)

of the rotation of three-dimensional space through an angle $2 \pi / 3$ about the line given in a rectangular coordinate system by the equations $x_{1}=x_{2}=x_{3}$ , in the basis of unit vectors of the coordinate axes;

(d)

of the projection of three-dimensional space onto the coordinate axis of the vector $e_{2}$ parallel to the coordinate plane of the vectors $e_{1}$ and $e_{3}$ , in the basis $(e_{1}, e_{2}, e_{3})$ ;

(e)

$x \mapsto (x, a) a$ in a Euclidean space, in the orthonormal basis $(e_{1}, e_{2}, e_{3})$ , where $a=e_{1}-2 e_{3}$ in the indicated basis;

(f)

$X \mapsto \left(\begin{array}{ll}a & b \\ c & d\end{array}\right) X$ on the space $\mathbf{M}_{2}(\mathbb {R})$ in the basis of matrix units;

(g)

$X \mapsto X\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ on the space $\mathbf{M}_{2}(\mathbb {R})$ in the basis of matrix units;

(h)

$X \mapsto { }^{t} X$ on the space $\mathbf{M}_{2}(\mathbb {R})$ in the basis of matrix units;

(i)

$X \mapsto A X B$ ( $A, B$ are fixed matrices) on the space $\mathrm{M}_{2}(\mathbb {R})$ in the basis consisting of matrix units;

(j)

$X \mapsto A X+X B(A, B-$ fixed matrices) on the space $\mathbf{M}_{2}(\mathbb {R})$ in the basis of matrix units;

(k)

of differentiation on the space $\mathbb {R}\left[x\right]_{n}$ in the basis $\left(1, x, \ldots , x^{n}\right)$ ;

(l)

of differentiation on the space $\mathbb {R}\left[x\right]_{n}$ in the basis $\left(x^{n}, x^{n-1}, \ldots , 1\right)$ ;

(m)

of differentiation on the space $\mathbb {R}\left[x\right]_{n}$ in the basis

\left(1, x-1, \frac{(x-1)^{2}}{2}, \ldots , \frac{(x-1)^{n}}{n!}\right)

Problem 39.16

Prove that in the space $\mathbb {R}^{3}$ there exists a unique linear operator taking the vectors $(1,1,1),(0,1,0)$ , $(1,0,2)$ respectively to the vectors $(1,1,1),(0,1,0),(1,0,1)$ , and find the matrix of this operator in the basis consisting of unit vectors.

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Problem 39.17

Let a vector space $V$ be the direct sum of subspaces $L_{1}$ and $L_{2}$ with bases $\left(a_{1}, \ldots , a_{k}\right)$ and $\left(a_{k+1}, \ldots , a_{n}\right)$ . Prove that the projection onto $L_{1}$ parallel to $L_{2}$ is a linear operator, and find its matrix in the basis $\left(a_{1}, \ldots , a_{n}\right)$ .

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Problem 39.18

Find the general form of the matrices of linear operators on an $n$ -dimensional space taking given linearly independent vectors $a_{1}, \ldots , a_{k}(k<n)$ to given vectors $b_{1}, \ldots , b_{k}$ , in a basis of the form $\left(a_{1}, \ldots , a_{k}, a_{k+1}, \ldots , a_{n}\right)$ .

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Problem 39.19

(a)(b)

Let a linear operator on a space $V$ have, in the basis $\left(e_{1}, \ldots , e_{4}\right)$ , the matrix

\left[\begin{smallmatrix} 0 & 1 & 2 & 3 \\ 5 & 4 & 0 & -1 \\ 3 & 2 & 0 & 3 \\ 6 & 1 & -1 & 7 \end{smallmatrix}\right]

Find the matrix of this operator in the bases:

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(a)

$\left(e_{2}, e_{1}, e_{3}, e_{4}\right)$ ;

(b)

$\left(e_{1}, e_{1}+e_{2}, e_{1}+e_{2}+e_{3}, e_{1}+e_{2}+e_{3}+e_{4}\right)$ .

Problem 39.20

Let a linear operator on the space $\mathbb {R}\left[x\right]_{2}$ have, in the basis $\left(1, x, x^{2}\right)$ , the matrix

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

Find its matrix in the basis

\left(3 x^{2}+2 x+1, x^{2}+3 x+2,2 x^{2}+x+3\right)

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Problem 39.21

Let a linear operator on the space $\mathbb {R}^{3}$ have, in the basis

((8,-6,7),(-16,7,-13),(9,-3,7))

the matrix

\left[\begin{smallmatrix} 1 & -18 & 15 \\ -1 & -22 & 20 \\ 1 & -25 & 22 \end{smallmatrix}\right]

Find its matrix in the basis

((1,-2,1),(3,-1,2),(2,1,2))

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Problem 39.22

Let a linear operator $\mathcal{A}$ on an $n$ -dimensional vector space $V$ take the linearly independent vectors $a_{1}, \ldots , a_{n}$ to the vectors $b_{1}, \ldots , b_{n}$ respectively. Prove that the matrix of this operator in some basis $\mathbf{e}=\left(e_{1}, \ldots , e_{n}\right)$ equals $B A^{-1}$ , where the columns of the matrices $A$ and $B$ consist respectively of the coordinates of the given vectors in the basis e.

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Problem 39.23

(a)(b)

Find the general form of the matrix of a linear operator $\mathcal{A}$ in a basis whose first $k$ vectors form:

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(a)

a basis of the kernel of the operator $\mathcal{A}$ ;

(b)

a basis of the image of the operator $\mathcal{A}$ .

Problem 39.24

Prove that if $f(t)=f_{1}(t) f_{2}(t)$ is a factorization of the polynomial $f(t)$ into coprime factors and the linear operator $\mathcal{A}$ satisfies $f(\mathcal{A})=0$ , then in some basis the matrix of the operator $\mathcal{A}$ has the form $\left(\begin{array}{cc}A_{1} & 0 \\ 0 & A_{2}\end{array}\right)$ , where $f_{1}\left(A_{1}\right)=0, f_{2}\left(A_{2}\right)=0$ .

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Section 40Eigenvectors, invariant subspaces, root subspaces[44/66%]

Problem 40.1

(a)(b)(c)(d)(e)(f)

Find the eigenvectors and eigenvalues:

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(a)

of the differentiation operator on the space $\mathbb {R}\left[x\right]_{n}$ ;

(b)

of the operator $X \mapsto { }^{t} X$ on the space $\mathbf{M}_{n}(\mathbb {R})$ ;

(c)

of the operator $x \frac{d}{d x}$ on the space $\mathbb {R}\left[x\right]_{n}$ ;

(d)

of the operator $\frac{1}{x} \int_{0}^{x} f(t) d t$ on the space $\mathbb {R}\left[x\right]_{n}$ ;

(e)

of the operator $f \mapsto \frac{d^{n} f}{d x^{n}}$ on the linear span

\langle 1, \cos x, \sin x, \ldots , \cos m x, \sin m x\rangle

(f)

of the operator $f \rightarrow \int_{-x}^{x} f(t) d t$ on the linear span

\langle \cos x, \sin x, \ldots , \cos m x, \sin m x\rangle

Problem 40.2

Prove that on the space $\mathbb {R}\left[x\right]_{n}$ the linear operator $f \mapsto f(a x+b)$ has the set of eigenvalues $1, a, \ldots , a^{n}$ .

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Problem 40.3

Prove that an eigenvector of a linear operator $\mathcal{A}$ with eigenvalue $\lambda$ is an eigenvector of the operator $f(\mathcal{A})$ , where $f(t)$ is a polynomial, with eigenvalue $f(\lambda )$ .

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Problem 40.4

Prove that if the operator $\mathcal{A}$ is nonsingular, then the operators $\mathcal{A}$ and $\mathcal{A}^{-1}$ have the same eigenvectors.

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Problem 40.5

Prove that all nonzero vectors of a space are eigenvectors of a linear operator $\mathcal{A}$ if and only if $\mathcal{A}$ is a similarity operator $x \mapsto \alpha x$ , where $\alpha$ is some fixed scalar.

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Problem 40.6

Prove that if a linear operator $\mathcal{A}$ on an $n$ -dimensional space has $n$ distinct eigenvalues, then any linear operator commuting with $\mathcal{A}$ has a basis consisting of its eigenvectors.

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Problem 40.7

Prove that the subspace $V_{\lambda }(\mathcal{A})$ , consisting of all eigenvectors of the operator $\mathcal{A}$ with eigenvalue $\lambda$ together with the zero vector, is invariant under any linear operator $\mathcal{B}$ commuting with $\mathcal{A}$ .

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Problem 40.8

(a)(b)

Prove that for any (possibly infinite) collection of commuting linear operators on a finite-dimensional complex space:

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(a)

there exists a common eigenvector;

(b)

there exists a basis in which the matrices of all these operators are upper triangular.

Problem 40.9

Prove that if the operator $\mathcal{A}^{2}$ has eigenvalue $\lambda^{2}$ , then one of the numbers $\lambda$ and $-\lambda$ is an eigenvalue of the operator $\mathcal{A}$ .

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Problem 40.10

(a)(b)

Prove that:

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(a)

the coefficients $c_{1}, \ldots , c_{n}$ of the polynomial

\left|A-\lambda E\right|=(-\lambda )^{n}+c_{1}(-\lambda )^{n-1}+\ldots +c_{n}

are sums of the principal minors of the corresponding order of the matrix $A$ ;

(b)

the sum and product of the characteristic numbers of the matrix $A$ equal its trace and determinant, respectively.

Problem 40.11

Prove that every polynomial of degree $n$ with leading coefficient $(-1)^{n}$ is the characteristic polynomial of some matrix of order $n$ .

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Problem 40.12

Prove that if $A$ and $B$ are square matrices of the same order, then the matrices $A B$ and $B A$ have the same characteristic polynomial.

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Problem 40.13

Find the characteristic numbers of the matrix ${ }^{t} A \cdot A$ , where $A$ is the row matrix $\left(a_{1}, \ldots , a_{n}\right)$ .

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Problem 40.14

Prove that all the characteristic numbers of a matrix are nonzero if and only if the matrix is nonsingular.

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Problem 40.15

(a)(b)(c)(d)(e)(f)

Find the eigenvalues and eigenvectors of the linear operators given in some basis by the matrices:

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(a)

$\left(\begin{array}{rrr}2 & -1 & 2 \\ 5 & -3 & 3 \\ -1 & 0 & -2\end{array}\right)$ ;

(b)

$\left(\begin{array}{rrr}0 & 1 & 0 \\ -4 & 4 & 0 \\ -2 & 1 & 2\end{array}\right) ;$

(c)

$\left(\begin{array}{lll}4 & -5 & 2 \\ 5 & -7 & 3 \\ 6 & -9 & 4\end{array}\right)$ ;

(d)

$\left(\begin{array}{rrr}7 & -12 & 6 \\ 10 & -19 & 10 \\ 12 & -24 & 13\end{array}\right)$ ;

(e)

$\left(\begin{array}{rrr}4 & -5 & 7 \\ 1 & -4 & 9 \\ -4 & 0 & 5\end{array}\right)$ ;

(f)

$\left(\begin{array}{rrrr}3 & -1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 3 & 0 & 5 & -3 \\ 4 & -1 & 3 & -1\end{array}\right)$ .

Problem 40.16

(a)(b)(c)(d)

Determine which of the following matrices can be reduced to diagonal form by passing to a new basis over the field $\mathbb {R}$ or over the field $\mathbb {C}$ :

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(a)

$\left(\begin{array}{rrr}-1 & 3 & -1 \\ -3 & 5 & -1 \\ -3 & 3 & 1\end{array}\right)$ ;

(b)

$\left(\begin{array}{rrr}4 & 7 & -5 \\ -4 & 5 & 0 \\ 1 & 9 & -4\end{array}\right)$ ;

(c)

$\left(\begin{array}{lll}4 & 2 & -5 \\ 6 & 4 & -9 \\ 5 & 3 & -7\end{array}\right)$ ;

(d)

$\left(\begin{array}{rrrr}1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & 1 & -1 \\ 1 & -1 & -1 & 1\end{array}\right)$ .

Find this basis and the corresponding form of the matrix.

Problem 40.17

Under what conditions is a matrix, all of whose entries except possibly the entries $\alpha_{1}, \ldots , \alpha_{n}$ of the antidiagonal are zero, similar to a diagonal matrix?

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Problem 40.18

For the matrix $A$ of order $n$ whose antidiagonal entries equal 1 and whose remaining entries equal zero, find a matrix $T$ such that the matrix $B=T^{-1} A T$ is diagonal. Compute the matrix $B$ .

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Problem 40.19

Prove that the number of linearly independent eigenvectors of a linear operator $\mathcal{A}$ with eigenvalue $\lambda$ is no greater than the multiplicity of $\lambda$ as a root of the characteristic polynomial of the operator $\mathcal{A}$ .

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Problem 40.20

Let $\lambda_{1}, \ldots , \lambda_{n}$ be the eigenvalues of a linear operator $\mathcal{A}$ on an $n$ -dimensional complex space. Find the eigenvalues of the operator $\mathcal{A}$ regarded as an operator on the corresponding $2 n$ -dimensional real space.

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Problem 40.21

(a)(b)

Let $\lambda_{1}, \ldots , \lambda_{n}$ be the roots of the characteristic polynomial of the matrix $A$ . Find the eigenvalues:

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(a)

of the linear operator $X \mapsto A X^{t} A$ on the space $\mathbf{M}_{n}(\mathbb {R})$ ;

(b)

of the linear operator $X \mapsto A X A^{-1}$ on the space $\mathbf{M}_{n}(\mathbb {R})$ (the matrix $A$ is nonsingular).

Problem 40.22

Find all invariant subspaces of the differentiation operator on the space $\mathbb {R}\left[x\right]_{n}$ .

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Problem 40.23

Prove that the linear span of any system of eigenvectors of a linear operator $\mathcal{A}$ is invariant under $\mathcal{A}$ .

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Problem 40.24

(a)(b)(c)(d)

Prove that:

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(a)

the kernel and image of a linear operator $\mathcal{A}$ are invariant under $\mathcal{A}$ ;

(b)

every subspace containing the image of the operator $\mathcal{A}$ is invariant under $\mathcal{A}$ ;

(c)

if a subspace $L$ is invariant under $\mathcal{A}$ , then its image and full preimage are invariant under $\mathcal{A}$ ;

(d)

if the linear operator $\mathcal{A}$ is nonsingular, then every subspace invariant under $\mathcal{A}$ is invariant under $\mathcal{A}^{-1}$ .

Problem 40.25

Prove that in an $n$ -dimensional complex space every linear operator has an invariant subspace of dimension $n-1$ .

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Problem 40.26

Prove that a linear operator on a vector space over a field $K$ that has, in some basis, the matrix

\left[\begin{smallmatrix} a_{1} & 1 & 0 & \ldots & 0 \\ a_{2} & 0 & 1 & \ldots & 0 \\ \ldots \ldots \ldots \ldots \ldots \ldots & \ldots \\ a_{n-1} & 0 & 0 & \ldots & 1 \\ a_{n} & 0 & 0 & \ldots & 0 \end{smallmatrix}\right]

where the polynomial $x^{n}-a_{1} x^{n-1}-\ldots -a_{n-1} x-a_{n}$ is irreducible over $K$ , has no nontrivial invariant subspaces.

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Problem 40.27

Let a linear operator $\mathcal{A}$ on an $n$ -dimensional space have, in some basis, a diagonal matrix with distinct entries on the diagonal. Find all subspaces invariant under $\mathcal{A}$ .

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Problem 40.28

Find all invariant subspaces of a linear operator that has, in some basis, a matrix consisting of a single Jordan block.

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Problem 40.29

Find all subspaces of a three-dimensional vector space that are invariant under the linear operator with matrix

\left[\begin{smallmatrix} 4 & -2 & 2 \\ 2 & 0 & 2 \\ -1 & 1 & 1 \end{smallmatrix}\right]

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Problem 40.30

Find all subspaces of a three-dimensional vector space that are invariant under two linear operators given by the matrices

\left[\begin{smallmatrix} 5 & -1 & -1 \\ -1 & 5 & -1 \\ -1 & -1 & 5 \end{smallmatrix}\right], \quad \left[\begin{smallmatrix} -6 & 2 & 3 \\ 2 & -3 & 6 \\ 3 & 6 & 2 \end{smallmatrix}\right]

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Problem 40.31

(a)(b)

In $\mathbb {R}\left[x\right]_{n}$ and $\mathbb {C}\left[x\right]_{n}$ , find all subspaces invariant under the operator:

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(a)

$\mathcal{A}(f)=x \frac{d f}{d x}$ ;

(b)

$\mathcal{A}(f)=\frac{1}{x} \int_{0}^{x} f(t) d t$ .

Problem 40.32

(a)(b)

In the linear span of the functions

\langle \cos x, \sin x, \ldots , \cos n x, \sin n x\rangle

find all subspaces invariant under the operator:

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(a)

$\mathcal{A}(f)=\frac{d f}{d x}$ ;

(b)

$\mathcal{A}(f)=\int_{-x}^{x} f(t) d t$ .

Problem 40.33

Prove that if the operators $\mathcal{A}, \mathcal{B}$ on a finite-dimensional vector space $V$ over the field $\mathbb {C}$ satisfy $\mathcal{A}^{2}=\mathcal{B}^{2}=\mathcal{E}$ , then $V$ contains a one- or two-dimensional subspace invariant under $\mathcal{A}$ and $\mathcal{B}$ .

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Problem 40.34

Prove that a complex vector space containing only one line invariant under a linear operator $\mathcal{A}$ is not decomposable into a direct sum of nonzero subspaces invariant under $\mathcal{A}$ .

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Problem 40.35

(a)(b)(c)(d)

Find the eigenvalues and root subspaces of the linear operator given in some basis by the matrix:

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(a)

$\left(\begin{array}{lll}4 & -5 & 2 \\ 5 & -7 & 3 \\ 6 & -9 & 4\end{array}\right)$ ;

(b)

$\left(\begin{array}{lll}1 & -3 & 4 \\ 4 & -7 & 8 \\ 6 & -7 & 7\end{array}\right)$ ;

(c)

$\left(\begin{array}{ccc}2 & 6 & -15 \\ 1 & 1 & -5 \\ 1 & 2 & -6\end{array}\right)$ ;

(d)

$\left(\begin{array}{rrrr}0 & -2 & 3 & 2 \\ 1 & 1 & -1 & -1 \\ 0 & 0 & 2 & 0 \\ 1 & -1 & 0 & 1\end{array}\right)$ .

Problem 40.36

Prove that a linear operator on a complex vector space has, in some basis, a diagonal matrix if and only if all its root vectors are eigenvectors.

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Problem 40.37

Prove that if a linear operator on a complex vector space has, in some basis, a diagonal matrix, then its restriction to any invariant subspace $L$ also has a diagonal matrix in some basis of the subspace $L$ .

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Problem 40.38

Prove that every root subspace of a linear operator $\mathcal{A}$ is invariant under any linear operator $\mathcal{B}$ commuting with $\mathcal{A}$ .

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Problem 40.39

Prove that if the matrix of a linear operator can be reduced to Jordan form, then every invariant subspace is the direct sum of its intersections with the root subspaces of the operator.

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Problem 40.40

Let $A \in \mathbf{M}_{n}(\mathbb {C})$ . Consider on the space $\mathbf{M}_{n \times m}(\mathbb {C})$ the operator $L_{A}$ , where $L_{A}(X)=A X$ . Find the eigenvalues of $L_{A}$ . Find the root subspaces of the operator $L_{A}$ , where $A$ is an upper triangular matrix.

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Problem 40.41

(a)(b)

Let $A \in \mathbf{M}_{n}(\mathbb {C}), B \in \mathbf{M}_{m}(\mathbb {C})$ , where $A$ and $B$ have no common eigenvalues. Prove that:

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(a)

if $X$ is a matrix of size $n \times m$ and $A X-X B=0$ , then $X=0$ ;

(b)

the equation $A X-X B=C$ , where $X, C$ are matrices of size $n \times m$ , has a solution, and moreover a unique one.

Problem 40.42

Let $\mathcal{A}$ be a linear operator on a finite-dimensional complex vector space $V$ . Prove that $V$ has a basis in which the matrix of the operator $\mathcal{A}$ is upper triangular.

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Problem 40.43

Let $\mathcal{A}$ be a linear operator on a finite-dimensional real space $V$ . Prove that $V$ has a basis in which the matrix of the operator $\mathcal{A}$ has the block upper-triangular form

\left[\begin{smallmatrix} \boxed {A_{1}} & & & * \\ & \boxed {A_{2}} & & \\ & & \ddots & \\ 0 & & & \boxed {A_{n}} \end{smallmatrix}\right],

where the square blocks $A_{1}, \ldots , A_{n}$ have size at most 2.

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Problem 40.44

Let $\mathcal{A}, \mathcal{B}$ be linear operators on a finite-dimensional complex vector space, and let the rank of the operator $\mathcal{A B}-\mathcal{B} \mathcal{A}$ not exceed 1. Prove that there exists a common eigenvector for $\mathcal{A}$ and $\mathcal{B}$ .

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Section 41Jordan canonical form and its applications. Minimal polynomial[47/66%]

Problem 41.1

(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)

Find the Jordan form of the matrix:

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(a)

$\left(\begin{array}{ccc}1 & -3 & 4 \\ 4 & -7 & 8 \\ 6 & -7 & 7\end{array}\right)$ ;

(b)

$\left(\begin{array}{rrr}4 & -5 & 7 \\ 1 & -4 & 9 \\ -4 & 0 & 5\end{array}\right)$ ;

(c)

$\left(\begin{array}{rrr}4 & 6 & 0 \\ -3 & -5 & 0 \\ -3 & -6 & 1\end{array}\right)$ ;

(d)

$\left(\begin{array}{rrr}3 & 0 & 8 \\ 3 & -1 & 6 \\ -2 & 0 & -5\end{array}\right)$ ;

(e)

$\left(\begin{array}{rrr}-2 & 8 & 6 \\ -4 & 10 & 6 \\ 4 & -8 & -4\end{array}\right) ; \quad$

(f)

$\left(\begin{array}{rrrr}1 & -3 & 0 & 3 \\ -2 & -6 & 0 & 13 \\ 0 & -3 & 1 & 3 \\ -1 & -4 & 0 & 8\end{array}\right)$ ;

(g)

$\left(\begin{array}{rrrr}3 & -1 & 1 & -7 \\ 9 & -3 & -7 & -1 \\ 0 & 0 & 4 & -8 \\ 0 & 0 & 2 & -4\end{array}\right)$ ;

(h)

$\left(\begin{array}{rrrrrrr}1 & -1 & 1 & 0 & \ldots & 0 & 0 \\ 0 & 1 & -1 & 0 & \ldots & 0 & 0 \\ 0 & 0 & 1 & -1 & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & 0 & \ldots & 1 & -1 \\ 0 & 0 & 0 & 0 & \ldots & 0 & 1\end{array}\right)$ ;

(i)

$\left(\begin{array}{ccccc}1 & 1 & 1 & \ldots & 1 \\ 0 & 1 & 1 & \ldots & 1 \\ 0 & 0 & 1 & \ldots & 1 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 1\end{array}\right) ; \quad$

(j)

$\left(\begin{array}{ccccc}n & n-1 & n-2 & \ldots & 1 \\ 0 & n & n-1 & \ldots & 2 \\ 0 & 0 & n & \ldots & 3 \\ \ldots & \ldots & \ldots & \ldots & \ldots \end{array}\right) ;$

(k)

$\left(\begin{array}{ccccc}1 & 0 & 0 & \ldots & 0 \\ 1 & 2 & 0 & \ldots & 0 \\ 1 & 2 & 3 & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 1 & 2 & 3 & \ldots & n\end{array}\right) ; \quad$

(l)

$\left(\begin{array}{cccccc}0 & 1 & 0 & 0 & \ldots & 0 \\ 0 & 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 0 & 1 & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & 0 & \ldots & 1 \\ 1 & 0 & 0 & 0 & \ldots & 0\end{array}\right)$ ;

(m)

$\left(\begin{array}{ccccc}\alpha & a_{12} & a_{13} & \ldots & a_{1 n} \\ 0 & \alpha & a_{23} & \ldots & a_{2 n} \\ 0 & 0 & \alpha & \ldots & a_{3 n} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & \alpha \end{array}\right)$ , where $a_{12}, a_{23}, \ldots , a_{n-1, n} \neq 0$ .

Problem 41.2

Prove that the Jordan form of the matrix $A+\alpha E$ equals $A_{j}+\alpha E$ , where $A_{j}$ is the Jordan form of the matrix $A$ .

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Problem 41.3

(a)(b)

Let $A$ be a Jordan block of order $n$ with entry $\alpha$ on the main diagonal.

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(a)

Find the matrix $f(A)$ , where $f(x)$ is a polynomial.

(b)

Find the Jordan form of the matrix $A^{2}$ .

Problem 41.4

Find the Jordan form of the matrix

\left[\begin{smallmatrix} \alpha & 0 & 1 & 0 & \ldots & 0 & 0 \\ 0 & \alpha & 0 & 1 & \ldots & 0 & 0 \\ 0 & 0 & \alpha & 0 & \ldots & 0 & 0 \\ 0 & 0 & 0 & \alpha & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & 0 & \ldots & \alpha & 0 \\ 0 & 0 & 0 & 0 & \ldots & 0 & \alpha \end{smallmatrix}\right].

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Problem 41.5

(a)(b)

Find the Jordan form of the matrix:

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(a)

$A^{2}$ ;

(b)

$A^{-1}(A$ is a nonsingular matrix); if $A$ has Jordan form $A_{j}$ .

Problem 41.6

(a)(b)

Find the Jordan form of the matrix $A$ and determine the geometric meaning of the corresponding linear operator $\mathcal{A}$ , if:

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(a)

$A^{2}=E$ ;

(b)

$A^{2}=A$ .

Problem 41.7

Prove that every periodic complex matrix is similar to a diagonal matrix, and find the form of this diagonal matrix.

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Problem 41.8

Prove that a matrix is nilpotent if and only if all its characteristic numbers equal zero.

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Problem 41.9

Prove that for every linear operator $\mathcal{A}$ of rank 1 on a complex vector space there exists a number $k$ such that $\mathcal{A}^{2}=k \mathcal{A}$ .

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Problem 41.10

(a)(b)(c)(d)

Find the Jordan form of the matrix of a linear operator $\mathcal{A}$ and a basis $(f_{1}, \ldots , f_{n})$ in which $\mathcal{A}$ has this matrix, if in the basis $\left(e_{1}, \ldots , e_{n}\right)$ the operator $\mathcal{A}$ is given by the matrix:

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(a)

$\left(\begin{array}{ccc}3 & 2 & -3 \\ 4 & 10 & -12 \\ 3 & 6 & -7\end{array}\right)$ ;

(b)

$\left(\begin{array}{rrr}1 & 1 & -1 \\ -3 & -3 & 3 \\ -2 & -2 & 2\end{array}\right)$ ;

(c)

$\left(\begin{array}{rrrr}0 & 1 & -1 & 1 \\ -1 & 2 & -1 & 1 \\ -1 & 1 & 1 & 0 \\ -1 & 1 & 0 & 1\end{array}\right)$ ;

(d)

$\left(\begin{array}{cccc}6 & -9 & 5 & 4 \\ 7 & -13 & 8 & 7 \\ 8 & -17 & 11 & 8 \\ 1 & -2 & 1 & 3\end{array}\right)$ .

Problem 41.11

Find the Jordan form of the matrix of a linear operator on a complex vector space having only one invariant line.

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Problem 41.12

Prove that the maximal number of linearly independent eigenvectors of a linear operator $\mathcal{A}$ with eigenvalue $\lambda$ equals the number of blocks with diagonal entry $\lambda$ in the Jordan form of the matrix of the operator $\mathcal{A}$ .

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Problem 41.13

Prove that the set of linear operators on an $n$ -dimensional complex vector space that commute with a given operator $\mathcal{A}$ is a vector space of dimension $\geqslant n$ .

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Problem 41.14

Prove that if a linear operator $\mathcal{B}$ on a complex vector space commutes with every linear operator that commutes with the operator $\mathcal{A}$ , then $\mathcal{B}$ is a polynomial in $\mathcal{A}$ .

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Problem 41.15

Prove that if matrices $A$ and $B$ satisfy the relation $A B-B A=B$ , then the matrix $B$ is nilpotent.

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Problem 41.16

On the space of complex polynomials of degree at most 2 in $x$ and $y$ , the operator $\mathcal{A}: f(x, y) \rightarrow f(x+1, y+1)$ acts. Find the Jordan form of the operator $\mathcal{A}$ .

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Problem 41.17

On the space of complex polynomials in $x, y$ of degree at most $n$ , the operator $\mathcal{A}=\frac{\partial }{\partial x}+\frac{\partial }{\partial y}$ acts. Find the Jordan form of $\mathcal{A}$ .

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Problem 41.18

Prove that any matrix is similar to its transpose.

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Problem 41.19

On the space $\mathbf{M}_{2}(\mathbb {C})$ consider the linear operator $L_{A}(X)=A X$ , where $X \in \mathbf{M}_{2}(\mathbb {C})$ and $A$ is a fixed matrix from $\mathbf{M}_{2}(\mathbb {C})$ . Find the Jordan form of the operator $L_{A}$ , given the Jordan form of $A$ .

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Problem 41.20

Prove that for any nonsingular square complex matrix $A$ and any natural number $k$ , the equation $X^{k}=A$ has a solution.

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Problem 41.21

(a)(b)

Solve the equations:

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(a)

$X^{2}=\left(\begin{array}{rr}3 & 1 \\ -1 & 5\end{array}\right)$ ;

(b)

$X^{2}=\left(\begin{array}{ll}6 & 2 \\ 3 & 7\end{array}\right)$ .

Problem 41.22

(a)(b)

Using Jordan form and problems 17.7-17.9, compute:

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(a)

$\left(\begin{array}{rr}1 & 1 \\ -1 & 3\end{array}\right)^{50}$ ;

(b)

$\left(\begin{array}{rr}7 & -4 \\ 14 & -8\end{array}\right)^{64}$ .

Problem 41.23

Find the minimal polynomial of a diagonal matrix with distinct entries on the main diagonal.

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Problem 41.24

Find the minimal polynomial of a Jordan block of order $n$ with the number $\alpha$ on the main diagonal.

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Problem 41.25

Prove that the minimal polynomial of a block-diagonal matrix equals the least common multiple of the minimal polynomials of its blocks.

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Problem 41.26

(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)

Find the minimal polynomial:

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(a)

of the identity operator;

(b)

of the zero operator;

(c)

of the projection operator of an $n$ -dimensional space $V$ onto a $k$ -dimensional subspace $L(0<k<n)$ ;

(d)

of a reflection operator;

(e)

of a nilpotent operator of index $k$ ;

(f)

of the operator $\mathcal{A}$ from 40.31(a);

(g)

of the operator $\mathcal{A}$ from 40.31(b);

(h)

of the operator $\mathcal{A}$ from 40.32(a);

(i)

of the operator $\mathcal{A}$ from 40.32(b);

(j)

of the operator $L_{A}$ from 40.40;

(k)

of the operator $\mathcal{A}$ from 41.16.

Problem 41.27

(a)(b)

Find the minimal polynomial of the matrix:

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(a)

$\left(\begin{array}{rrr}3 & -1 & -1 \\ 0 & 2 & 0 \\ 1 & 1 & 1\end{array}\right)$ ;

(b)

$\left(\begin{array}{rrr}4 & -2 & 2 \\ -5 & 7 & -5 \\ -6 & 6 & -4\end{array}\right)$ .

Problem 41.28

Let a linear operator $\mathcal{A}$ have, in the basis $(e_{1}, e_{2}, e_{3})$ of the space $V$ , the matrix

\left[\begin{smallmatrix} 1 & 0 & 0 \\ 1 & 2 & 1 \\ -1 & 0 & 1 \end{smallmatrix}\right]

Find the minimal polynomial $g(t)$ of the operator $\mathcal{A}$ and decompose the space $V$ into a direct sum of invariant subspaces corresponding to the factorization of the minimal polynomial into coprime factors.

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Problem 41.29

Prove that the minimal polynomial of a matrix of order $\geqslant 2$ and rank 1 has degree 2.

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Problem 41.30

What can be said about the Jordan form of the matrix of a linear operator $\mathcal{A}$ on a complex space if $\mathcal{A}^{3}=\mathcal{A}^{2}$ ?

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Problem 41.31

Prove that some power of the minimal polynomial of a matrix is divisible by its characteristic polynomial.

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Problem 41.32

Prove that for two matrices to be similar it is necessary, but not sufficient, that they have the same characteristic and minimal polynomials.

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Problem 41.33

Prove that if the degree of the minimal polynomial of a linear operator $\mathcal{A}$ equals the dimension of the space, then every operator commuting with $\mathcal{A}$ is a polynomial in $\mathcal{A}$ .

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Problem 41.34

(a)(b)(c)

A linear operator is called semisimple if every invariant subspace has an invariant complementary subspace.

Prove that:

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(a)

the restriction of a semisimple operator to an invariant subspace is also a semisimple operator;

(b)

a linear operator is semisimple if and only if the space is a direct sum of minimal invariant subspaces;

(c)

if for a linear operator $\mathcal{A}$ there exists a decomposition of the space into a direct sum of invariant subspaces, on each of which the restriction of the operator $\mathcal{A}$ is semisimple, then $\mathcal{A}$ is semisimple on the whole space.

Problem 41.35

Prove that if the minimal polynomial of a linear operator $\mathcal{A}$ in a space $V$ is a product of coprime polynomials $g_{1}(x)$ and $g_{2}(x)$ , then $V$ can be decomposed into a direct sum of two invariant subspaces such that the restrictions of the operator $\mathcal{A}$ to these subspaces have minimal polynomials $g_{1}(x)$ and $g_{2}(x)$ respectively.

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Problem 41.36

Prove that for any linear operator there exists a decomposition of the space into a direct sum of invariant subspaces such that the minimal polynomials of its restrictions to these subspaces are powers of distinct irreducible polynomials.

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Problem 41.37

Prove that if the minimal polynomial of a linear operator $\mathcal{A}$ is an irreducible polynomial of degree $k$ , then for any $x \neq 0$ the vectors $x, \mathcal{A} x, \ldots , \mathcal{A}^{k-1} x$ form a basis of the minimal invariant subspace.

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Problem 41.38

Prove that a linear operator is semisimple if and only if its minimal polynomial has no repeated irreducible factors.

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Problem 41.39

Prove that a linear operator in a vector space over a field $K$ of characteristic 0 is semisimple if and only if it has an eigenbasis over some extension of the field $K$ .

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Problem 41.40

Prove that the sum of two commuting semisimple linear operators over a field of characteristic 0 is a semisimple operator.

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Problem 41.41

(a)(b)(c)

Let $\mathcal{A}$ be a linear operator in a vector space over a field $K$ of characteristic 0, and let $K[\mathcal{A}]$ be the ring of linear operators representable as polynomials in $\mathcal{A}$ . Prove that if the minimal polynomial of the operator $\mathcal{A}$ is a power of an irreducible polynomial $p(x)$ , then:

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(a)

the elements of the ring $K[\mathcal{A}]$ divisible in this ring by the element $p(\mathcal{A})$ form an ideal $I$ distinct from $K[\mathcal{A}]$ ;

(b)

for every operator $\mathcal{B} \in I$ the minimal polynomial of the operator $\mathcal{A}+\mathcal{B}$ is divisible by $p(x) ;$

(c)

there exists an operator $\mathcal{B} \in I$ such that the minimal polynomial of the operator $\mathcal{A}+\mathcal{B}$ equals $p(x)$ .

Problem 41.42

Prove that every linear operator $\mathcal{A}$ in a vector space over a field of characteristic 0 can be represented as a sum of a semisimple and a nilpotent operator, both of which are polynomials in $\mathcal{A}$ .

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Problem 41.43

Prove that every linear operator $\mathcal{A}$ can be uniquely represented as a sum of commuting semisimple and nilpotent operators.

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Problem 41.44

(a)(b)

Prove that if the degree of the minimal polynomial $g(x)$ of a linear operator $\mathcal{A}$ in a vector space $V$ over a field $K$ equals the dimension of the space and $g(t)$ is a power of a polynomial irreducible over $K$ , then:

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(a)

$V$ cannot be decomposed into a direct sum of two invariant subspaces;

(b)

$V$ is cyclic with respect to $\mathcal{A}$ .

Problem 41.45

(a)(b)(c)

Let $\lambda_{1}, \ldots , \lambda_{n}$ be the eigenvalues of a matrix $A \in \mathbf{M}_{n}(\mathbb {C})$ .

Prove that:

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(a)

for any natural number $k$

\operatorname {tr} A^{k}=\lambda _{1}^{k}+\ldots +\lambda _{n}^{k}

(b)

the coefficients of the characteristic polynomial of the matrix $A$ are polynomials in $\operatorname {tr} A, \ldots , \operatorname {tr} A^{n}$ ;

(c)

if $\operatorname {tr} A=\operatorname {tr} A^{2}=\ldots =\operatorname {tr} A^{n}=0$ , then the matrix $A$ is nilpotent.

Problem 41.46

Let

B=\left[\begin{smallmatrix} 0 & 0 & \ldots & 0 & 1 \\ 0 & 0 & \ldots & 1 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 1 & \ldots & 0 & 0 \\ 1 & 0 & \ldots & 0 & 0 \end{smallmatrix}\right] \in \mathbf{M}_{n}(\mathbb {C})

and $S=\frac{1}{\sqrt{2}}(E+i B)$ . Prove that for the Jordan block $J(n, \lambda ) \in \mathbf{M}_{n}(\mathbb {C})$ the following equality holds $S J(n, \lambda ) S^{-1}=\lambda E+\frac{1}{2} \sum_{k=1}^{n-1}\left[E_{k, k+1}+E_{k+1, k}+i\left(E_{n-k+1, k+1}-E_{n-k, k}\right)\right]$ .

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Problem 41.47

Prove that every complex matrix is similar to a symmetric matrix.

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Section 42Normed vector spaces and algebras. Non-negative matrices[34/29%]

Problem 42.1

(a)(b)(c)

Let $K$ be a normed field (see 66.35) with norm $\left|x\right|$ . Prove that the following functions on $K^{n}$ are norms:

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(a)

$\left\| \left(a_{1}, \ldots , a_{n}\right)\right\| =\left|a_{1}\right|+\ldots +\left|a_{n}\right|$ ;

(b)

$\left\| \left(a_{1}, \ldots , a_{n}\right)\right\| =\max \left(\left|a_{1}\right|, \ldots ,\left|a_{n}\right|\right)$ ;

(c)

$\left\| \left(a_{1}, \ldots , a_{n}\right)\right\| =\sqrt{\left|a_{1}\right|^{2}+\ldots +\left|a_{n}\right|^{2}}$ .

Problem 42.2

Let $K$ be a locally compact normed field, $V$ a finite-dimensional vector space over $K$ . Prove that any two norms $\left\| x\right\|_{1},\left\| x\right\|_{2}$ on $V$ are equivalent, i.e., there exist positive real numbers $C_{1}, C_{2}$ such that for all $x \in V$

C_{1}\left\| x\right\| _{1} \leqslant \left\| x\right\| _{2} \leqslant C_{2}\left\| x\right\| _{1}

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Problem 42.3

Let $K$ be a normed field, $V$ a finite-dimensional vector space with basis $\left(e_{1}, \ldots , e_{m}\right)$ . Let $x_{n}=x_{n 1} e_{1}+\ldots +x_{n m} e_{m} \in V, x_{i j} \in K, n \geqslant 1$ . Prove that the sequence of vectors $x_{n}$ converges if and only if the sequences $x_{n i}$ converge for $i=1, \ldots , m$ .

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Problem 42.4

Let $K$ be a complete normed field and $V$ a finite-dimensional normed vector space over $K$ . Prove that $V$ is complete.

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Problem 42.5

(a)(b)(c)

Let $K$ be a normed field and $V$ a normed vector space over $K$ . Denote by $L(V)$ the set of all linear operators $\mathcal{A}$ in $V$ such that the number $\left\| \mathcal{A}(x)\right\|$ is bounded whenever $\left\| x\right\| =1$ . Prove that:

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(a)

$L(V)$ is a subspace of the space of all linear operators in $V$ ;

(b)

$L(V)$ is a normed algebra with norm

\left\| \mathcal{A}\right\| =\sup _{\left\| x\right\| =1}\left\| \mathcal{A}(x)\right\| ;

(c)

if $V$ is finite-dimensional, then $L(V)$ is the space of all operators in $V$ .

Problem 42.6

(a)(b)

Let $K$ be a normed field and let the norms (a), (b) from problem 42.1 be given on $K^{n}$ . Prove that (in accordance with problem 42.5) the norms in $\mathbf{M}_{n}(K)=L(V)$ have the form:

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(a)

$\left\| A\right\| =\max_{1 \leqslant j \leqslant n}\left(\sum_{i=1}^{n}\left|a_{i j}\right|\right)$ ;

(b)

$\left\| A\right\| =\max_{1 \leqslant i \leqslant n}\left(\sum_{j=1}^{n}\left|a_{i j}\right|\right)$ .

Problem 42.7

(a)(b)(c)

Let $K$ be a normed field. For $A=\left(a_{i j}\right) \in \mathbf{M}_{n}(K)$ set:

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(a)

$\left\| A\right\|_{1}=\sum_{i, j=1}^{n}\left|a_{i j}\right|$ ;

(b)

$\left\| A\right\|_{2}=\sqrt{\sum_{i, j=1}^{n}\left|a_{i j}\right|^{2}}$ ;

(c)

$\left\| A\right\|_{3}=n \cdot \max_{1 \leqslant i, j \leqslant n}\left|a_{i j}\right|$ .

Prove that each of these functions endows $\mathrm{M}_{n}(K)$ with the structure of a normed algebra.

Problem 42.8

Prove that for any matrix $A \in \mathbf{M}_{n}(\mathbb {C})$ the matrices $e^{A}, \sin A, \cos A$ are well defined.

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Problem 42.9

(a)(b)(c)(d)

Let $A \in \mathbf{M}_{n}(\mathbb {C})$ . Prove that:

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(a)

$\sin 2 A=2 \sin A \cos A$ ;

(b)

$e^{i A}=\cos A+i \sin A$ ;

(c)

$\sin A=\frac{1}{2}\left(e^{i A}-e^{-i A}\right)$ ;

(d)

$\cos A=\frac{1}{2}\left(e^{i A}+e^{-i A}\right)$ .

Problem 42.10

If $A, B \in \mathbf{M}_{n}(\mathbb {C})$ and $A B=B A$ , then $e^{A+B}=e^{A} e^{B}$ .

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Problem 42.11

Let $A$ be a normed algebra over a complete normed field $K$ . Prove that if $x \in A$ , then the limit exists

\rho (x)=\lim _{n \rightarrow \infty }\left\| x^{n}\right\| ^{1 / n}

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Problem 42.12

Let $x$ be an element of a Banach algebra $A$ over a complete normed field $K$ . Prove that the radius of convergence of the series $\sum_{n \geqslant 0} t^{n} x^{n} \in A[[t]]$ equals $\rho (x)^{-1}$ (see problem 42.11).

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Problem 42.13

(a)(b)

Let $x$ be an element of a Banach algebra $A$ over a complete normed field $K$ . The spectrum $\operatorname {Sp}(x)$ is the set of all $\lambda \in K$ such that the element $x-\lambda$ is not invertible in $A$ .

Prove that:

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(a)

the radius of the smallest disk in $K$ centered at zero containing $\operatorname {Sp}(x)$ equals $\rho (x)$ ;

(b)

the set $\operatorname {Sp}(x)$ is compact in $K$ .

Problem 42.14

Let $A \in \mathbf{M}_{n}(\mathbb {C})$ and let $\lambda_{1}, \ldots , \lambda_{n}$ be all the eigenvalues of the matrix $A$ . Prove that

\max _{1 \leqslant i \leqslant n}\left|\lambda _{i}\right|=\rho (A) \leqslant \left\| A\right\|

where $\left\| A\right\|$ is an arbitrary norm on the algebra $\mathbf{M}_{n}(\mathbb {C})$ , induced by virtue of 42.5(b) by some norm on $\mathbb {C}^{n}$ .

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Problem 42.15

Let $x$ be an element of a Banach algebra $A$ over a complete normed field $K$ and $f(t) \in K[[t]]$ . Prove that if $\rho (x)$ is less than the radius of convergence of $f(t)$ , then the series $f(x)$ converges.

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Problem 42.16

Let $x$ be an element of a Banach algebra $A$ over a complete normed field $K, g$ an invertible element of $A$ , and $f(t) \in K[[t]]$ . Prove that the series $f(x)$ converges if and only if $f\left(g x g^{-1}\right)$ converges, in which case $f\left(g x g^{-1}\right)=g f(x) g^{-1}$ .

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Problem 42.17

Let $a$ be an element of a Banach algebra over a complete normed field $K$ . Suppose that $f(t) \in K[[t]]$ and the series $f(a)$ converges. Suppose $a$ has an annihilating polynomial of degree $n$ . Prove that $f(a) \in \left\langle 1, a, a^{2}, \ldots , a^{n-1}\right\rangle$ .

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Problem 42.18

Let $A \in \mathbf{M}_{n}(\mathbb {C})$ and $\lambda_{0}, \ldots , \lambda_{n}$ be all the roots of the characteristic polynomial of the matrix $A$ with multiplicities $k_{0}, \ldots , k_{n}$ . Suppose that $\lambda_{0}, \ldots , \lambda_{n}$ lie inside the disk of convergence of the series $u(t) \in \mathbb {C}\left[\left[t\right]\right]$ .

Prove that

u(A)=\sum _{i=0}^{n} G_{i}(A) \sum _{k=0}^{k_{i}} \sum _{l=0}^{k} \frac{1}{l!}\left[\left.\frac{d^{l} u}{d x^{l}}\left(\lambda _{i}\right) \cdot \frac{d^{l}}{d x^{l}} G_{i}(x)^{-1}\right|_{x=\lambda _{i}}\left(A-\lambda _{i} E\right)^{k}\right]

where $G_{i}(t)=\prod_{j \neq i}\left(t-\lambda_{j}\right)^{k_{j}}$ .

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Problem 42.19

(a)(b)(c)(d)(e)

Compute:

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(a)

$\exp \left(\begin{array}{rr}3 & -1 \\ 1 & 1\end{array}\right)$ ;

(b)

$\exp \left(\begin{array}{ll}4 & -2 \\ 6 & -3\end{array}\right)$ ;

(c)

$\exp \left(\begin{array}{rrr}4 & 2 & -5 \\ 6 & 4 & -9 \\ 5 & 3 & 7\end{array}\right)$ ;

(d)

$\ln \left(\begin{array}{ccc}4 & -15 & 6 \\ 1 & -4 & 2 \\ 1 & -5 & 3\end{array}\right)$ ;

(e)

$\sin \left(\begin{array}{cc}\pi -1 & 1 \\ -1 & \pi +1\end{array}\right)$ .

Problem 42.20

Find the determinant of the matrix $e^{A}$ , where $A$ is a square matrix of order $n$ .

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Problem 42.21

Let $A \in \mathbf{M}_{n}(\mathbb {C})$ and

x(t)=\left[\begin{smallmatrix} x_{1}(t) \\ \vdots \\ x_{n}(t) \end{smallmatrix}\right]

a continuously differentiable vector function. Prove that the solution of the system of differential equations with constant coefficients

\frac{d}{d t} x(t)=A x(t)

with initial condition $x\left(t_{0}\right)$ is the vector function $x(t)=e^{A t} x\left(t_{0}\right)$ .

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Problem 42.22

Let $A$ be a nonnegative matrix and let $A^{k}$ be a positive matrix for some natural number $k$ . Prove that $\rho (A)>0$ .

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Problem 42.23

Give an example of a nonnegative $2 \times 2$ matrix that is not positive, such that $A^{2}$ is a positive matrix.

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Problem 42.24

Let a nonnegative matrix $A$ have a positive eigenvector. Prove that the matrix $A$ is similar to a nonnegative matrix in which the sums of the entries of each row are equal.

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Problem 42.25

Let $A, B$ be positive matrices, with the matrix $A-B$ also positive. Prove that $\rho (A)>\rho (B)$ .

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Problem 42.26

Let $A$ be a nonsingular nonnegative matrix such that the inverse matrix $A^{-1}$ is also nonnegative. Prove that $A=D P$ , where $D$ is a diagonal nonnegative invertible matrix and $P$ is a permutation matrix.

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Problem 42.27

Let $A$ be a nonnegative matrix and let $x$ be a nonzero complex vector such that $A x-\alpha x$ is a nonnegative vector for some real number $\alpha$ . Prove that $\rho (A)>\alpha$ .

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Problem 42.28

Let $A$ be a nonnegative matrix such that ${ }^{t} A$ has a positive eigenvector. Prove that if $A x-\rho (A) x$ is a nonnegative vector for some nonzero vector $x$ , then $A x=\rho (A) x$ .

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Problem 42.29

(a)(b)

Let $A$ be a nonnegative matrix such that the matrix $A^{k}$ is positive for some natural number $k$ . Prove that:

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(a)

$A$ has a positive eigenvector;

(b)

$\rho (A)$ is an eigenvalue of $A$ of multiplicity 1.

Problem 42.30

Let a nonnegative matrix $A$ have an eigenvector $x=\left(x_{1}, \ldots , x_{n}\right)$ , where $x_{1}, \ldots , x_{r}>0, x_{r+1}=\ldots =x_{n}=0$ . Prove that then there exists a permutation matrix $P$ such that $P^{-1} A P=\left(\begin{array}{cc}B & C \\ 0 & D\end{array}\right)$ , where $B \in \mathbf{M}_{r}(\mathbb {R}), D \in \mathbf{M}_{n-r}(\mathbb {R})$ , and $B$ has a positive eigenvector.

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Problem 42.31

Let $A$ be a nonnegative matrix. Prove that there exists a positive matrix $B$ commuting with $A$ if and only if the matrices $A$ and ${ }^{t} A$ have positive eigenvectors.

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Problem 42.32

Let $A$ be a nonnegative tridiagonal matrix. Prove that all eigenvalues of $A$ are real.

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Problem 42.33

Let $A$ be a nonnegative matrix. Prove that $\rho (E+A)=1+\rho (A)$ .

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Problem 42.34

(a)(b)(c)(d)

For the nonnegative matrices $A$ find the nonnegative eigenvectors $x$ and $\rho (A)$ :

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(a)

$\left(\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right)$ ;

(b)

$\left(\begin{array}{ll}3 & 4 \\ 5 & 2\end{array}\right)$ ;

(c)

$\left(\begin{array}{lll}1 & 4 & 8 \\ 0 & 2 & 6 \\ 1 & 0 & 0\end{array}\right)$ ;

(d)

$\left(\begin{array}{llll}3 & 2 & 3 & 1 \\ 0 & 2 & 0 & 5 \\ 5 & 7 & 1 & 6 \\ 0 & 1 & 0 & 4\end{array}\right)$ .