Eigenvalues, Eigenvectors

Prove that the set of all eigenvectors of the operator А with one and the same eigenvalue $\lambda$, together with the zero vector, forms a vector subspace. This subspace is called the eigenspace corresponding to the eigenvalue $\lambda$, and is denoted by $V_{\lambda }(\mathbf{A})$.

Prove that the set of eigenvalues of the operator A on a finite-dimensional space coincides with the set of roots of the polynomial $\chi_{\mathrm{A}}(\lambda )$. The multiplicity of the root $\lambda$ in the characteristic polynomial will be called the multiplicity of the eigenvalue $\lambda$.

Prove that if $v_{1}, \ldots , v_{k}$ are eigenvectors of one and the same operator with pairwise distinct eigenvalues $\lambda_{1}, \ldots , \lambda_{k}$, then the vectors $v_{1}, \ldots , v_{k}$ are linearly independent.

Let $V$ be a real space and let $\mathrm{J}, \mathrm{J}^{2} = -\mathrm{Id}$, be a complex structure on it. Prove that:

Let $V$ be a complex space. The conjugate complex vector space $\bar{V}$ coincides with $V$ as a set, the operation of vector addition also coincides with addition in $V$, while multiplication by complex numbers is defined as follows:

(on the right-hand side of the last equality, multiplication in $V$ is meant). Establish a natural isomorphism $\left(V_{\mathbb {R}}\right)_{\mathbb {C}} \rightarrow V \oplus \bar{V}$.

Prove that if the characteristic polynomial $\chi_{\mathrm{A}}(\lambda )$ of the operator А, acting in a space $V$ of dimension $n$, has exactly $n$ pairwise distinct roots, then there exists a basis $e_{1}, \ldots , e_{n}$ in which the matrix of the operator А is diagonal.

Prove that all eigenvalues of the operator А on a finite-dimensional space are nonzero if and only if $\det A \neq 0$.

Prove that if А is a nonsingular operator on a finite-dimensional space, then А and $\mathbf{A}^{-1}$ have the same eigenvectors. How are their corresponding eigenvalues related to each other?

Prove that the dimension of the eigenspace $V_{\lambda }(\mathbf{A})$ does not exceed the multiplicity of the root $\lambda$ in the characteristic polynomial of the operator A.

Prove that the multiplicity of the root $\lambda$ of a semisimple operator coincides with the dimension of the eigenspace $V_{\lambda }$ of this operator.

Find the eigenvalues, their multiplicities, and the eigenspaces of the linear operators:

Find the eigenvalues and eigenspaces of the transposition operator $\mathbf{T}: X \rightarrow X^{\top }$, acting in the space Mat $_{n}$ of square matrices of order $n$.

Find the eigenvalues and eigenspaces of the operator:

Find the eigenvalues, their multiplicities, and the corresponding eigenspaces of the differentiation operator $\frac{d}{d x}$ and the double-differentiation operator $\frac{d^{2}}{d x^{2}}$, acting:

Let $p_{0}(t)$ be an arbitrary fixed real polynomial, not a constant. Define a transformation $\varphi$ of the space $\mathbb {R}^{t}$ of all real polynomials into itself, which assigns to an arbitrary polynomial $p(t)$ its remainder $r(t)$ upon division by $p_{0}(t): \varphi (p(t)) = r(t), p(t) = q(t) p_{0}(t)+r(t)$.

Let А, В be linear operators on a finite-dimensional space. Prove that the eigenvalues of the operators АВ and ВА coincide.

Let А, В be linear operators on an arbitrary vector space. Prove that the nonzero eigenvalues of the operators АВ and ВА coincide.

Let $x$ be an eigenvector of the operator А with eigenvalue $\lambda$, and let $f(t)$ be some polynomial. Prove that $x$ is an eigenvector of the operator $f(\mathbf{A})$ with eigenvalue $f(\lambda )$.

Let D be the differentiation operator, and let $\mathbf{X}$ be the operator of multiplication by $x$ on the infinite-dimensional space $\mathbb {R}^{x}$ of all polynomials (or the space $C^{\infty }(\mathbb {R})$ of infinitely smooth functions) in $x$ with real coefficients. Also consider the operators

Prove the following identities:

Under the conditions of the previous problem, prove that if the function $\psi (x)$ is an eigenvector of the operator $\mathbf{H}$ with eigenvalue $\lambda$, then $\mathbf{A}_{+} \psi$ and $\mathbf{A}_{-} \psi$ are also eigenvectors (if they are nonzero) of the operator $\mathbf{H}$ with eigenvalues $\lambda +2$ and $\lambda -2$ respectively.

Problem 1172: let D be the differentiation operator, and let $\mathbf{X}$ be the operator of multiplication by $x$ on the infinite-dimensional space $\mathbb {R}^{x}$ of all polynomials (or the space $C^{\infty }(\mathbb {R})$ of infinitely smooth functions) in $x$ with real coefficients. Also consider the operators

Under the conditions of problem 1172, prove that if $\psi_{0}$ lies in the kernel of the operator $\mathbf{A}_{-}$, then the vectors $\psi_{k} = \mathbf{A}_{+}^{k} \psi_{0}$ are eigenvectors of the operator Н. Compute the corresponding eigenvalues $\lambda_{k}$.

Problem 1172: let D be the differentiation operator, and let $\mathbf{X}$ be the operator of multiplication by $x$ on the infinite-dimensional space $\mathbb {R}^{x}$ of all polynomials (or the space $C^{\infty }(\mathbb {R})$ of infinitely smooth functions) in $x$ with real coefficients. Also consider the operators

Consider

(Hermite polynomials), where the operator $\mathbf{A}_{+}$ is defined in problem 1172. Prove the following identities:

Prove that for a given operator А on an $n$-dimensional space over a field $\mathbb {k}$, there exists a basis in which А has a diagonal matrix if and only if the following two conditions hold:

Determine which of the following matrices of linear operators can be reduced to diagonal form by a change of basis. In the case where this is possible, find a basis of eigenvectors and the matrix of the operator in this basis:

The matrices $A$ and $B$ are called similar if there exists a nonsingular matrix $C$ for which $B = C^{-1} A C$. Determine which of the two matrices

is similar to the matrix $\left(\begin{array}{rrr}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{array}\right)$.

Determine which of the two matrices

is similar to the matrix

Prove that if the operators А and В commute, then for any eigenvalue $\lambda$ of the operator А the eigenspace $V_{\lambda }(\mathbf{A})$ is invariant under В.

Prove that a linear operator on a finite-dimensional space over an algebraically closed field always has at least one eigenvector.

Prove that a linear operator on a space $V$ of dimension $n$ over an algebraically closed field always has an invariant subspace of dimension $n-1$.

Prove that for any linear operator А on a finite-dimensional space over an algebraically closed field there exists a basis in which the matrix of this operator is upper triangular.

Reduce the matrices of the transformation to triangular form and find a basis in which they have this form:

Show that in a space $V$ of dimension $n$ over an algebraically closed field every operator А has invariant subspaces of all dimensions from 0 to $n$.

Let $\mathrm{A}_{1}, \ldots , \mathrm{A}_{k}, \ldots$ be a collection of pairwise commuting operators on a finite-dimensional space over an algebraically closed field. Prove that