Jordan canonical form and its applications. Minimal polynomial

Find the Jordan form of the matrix:

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 $A$.

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

Find the Jordan form of the matrix

Find the Jordan form of the matrix:

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

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

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

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

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:

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

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 $\mathcal{A}$.

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 $\geqslant n$.

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 $\mathcal{A}$.

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

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 $\mathcal{A}$.

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 $\mathcal{A}$.

Prove that any matrix is similar to its transpose.

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 $A$.

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

Solve the equations:

Using Jordan form and problems 17.7-17.9, compute:

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

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

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

Find the minimal polynomial:

Find the minimal polynomial of the matrix:

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

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.

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

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

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

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

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 $\mathcal{A}$.

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

Prove that:

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.

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.

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.

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

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 $K$.

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

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:

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 $\mathcal{A}$.

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

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:

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

Prove that:

Let

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 $S J(n, \lambda ) S^{-1}=\lambda E+\frac{1}{2} \sum_{k=1}^{n-1}\left[E_{k, k+1}+E_{k+1, k}+i\left(E_{n-k+1, k+1}-E_{n-k, k}\right)\right]$.

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