Substituting a Linear Operator into a Polynomial. Annihilating Polynomials

Prove that for any two polynomials $P(x)$, $Q(x)$ and any linear operator $\mathbf{A}$, the operators $P(\mathbf{A})$ and $Q(\mathbf{A})$ commute with each other.

Prove that any annihilating polynomial of an operator is divisible by its minimal polynomial. Deduce from this that if an operator has at least one annihilating polynomial, then its minimal polynomial is uniquely determined.

Find the minimal polynomial of the operator А on the space $V$ in each of the following cases:

Prove that the minimal polynomial of an operator does not change under a field extension. Namely, let А, А' be operators on finite-dimensional spaces over the fields $\mathbb {k}$ and $\mathbb {k}^{\prime }$ respectively, where $\mathbb {k}$ is a subfield of $\mathbb {k}^{\prime }$. Prove that if the matrices of the operators А, А' coincide, then their minimal polynomials also coincide. Generalize this statement to the case of arbitrary (not necessarily finite-dimensional) spaces.

Let А be an operator on a finite-dimensional space $V$, and let $U$ be some subspace of $V$ invariant under А. Let $\mathbf{A}^{\prime }: U \rightarrow U$ be the restriction of the operator А to the subspace $U$, and let $\mathbf{A}^{\prime \prime }: V / U \rightarrow V / U$ be the operator on the quotient space defined by the formula

Prove that

Let the polynomial $f(\lambda )$ annihilate the operator А acting in the space $V$, where $f(\lambda ) = f_{1}(\lambda ) f_{2}(\lambda )$, and the polynomials $f_{1}(\lambda )$ and $f_{2}(\lambda )$ are coprime. Prove that the space $V$ can be decomposed into a direct sum $V = L_{1} \oplus L_{2}$ of two subspaces invariant under А.

Prove that for any coprime polynomials $P(x), Q(x)$ and any linear operator А, the following equality holds

Generalize the previous problem to the case of an arbitrary number of pairwise coprime polynomials.

Call a polynomial $S(x)$ semisimple if, for every factorization $S(x) = Q(x) R(x)$ into a product of polynomials, the factors $Q(x)$ and $R(x)$ are coprime. Prove that the minimal polynomial $\mu_{\mathrm{A}}(\lambda )$ of the operator $\mathbf{A}$ on a finite-dimensional vector space is divisible by the maximal semisimple divisor of its characteristic polynomial $\chi_{\mathrm{A}}(\lambda )$.

Prove that a linear operator on a finite-dimensional space is semisimple if and only if its minimal polynomial $\mu_{\mathrm{A}}(\lambda )$ is semisimple (and hence coincides with the maximal semisimple divisor of the characteristic polynomial $\chi_{\mathrm{A}}(\lambda )$).

Prove that if an operator А is simultaneously semisimple and nilpotent, then it is the zero operator.

Show that the matrix $\left(\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right)$ defines:

1. a semisimple operator in the standard basis of the space $\mathbb {R}^{2}$;

2. a nilpotent operator in the standard basis of a two-dimensional space over $\mathbb {Z}_{2}$.

Which of the following operators А on the space $V$ are nilpotent, and which are semisimple:

(Jordan decomposition). Prove that any operator А on a finite-dimensional vector space over a field of characteristic zero can be represented as a sum $\mathbf{A} = \mathbf{S}+\mathbf{N}$ of commuting operators $\mathbf{S}$ and $\mathbf{N}$, in which $\mathbf{S}$ is semisimple and $\mathbf{N}$ is nilpotent.

For each of the following operators, given by its matrix in the standard basis of the space $\mathbb {k}^{n}$, find the Jordan decomposition:

Let $V$ be the space of all infinite sequences $\left(x_{1}, x_{2}, \ldots \right)$ with elements from $\mathbb {k}$, and let $P(\lambda ) = \lambda^{n}-a_{n-1} \lambda^{n-1}-\ldots -a_{1} \lambda -a_{0}$ be some polynomial. Consider the shift operator $\mathbf{T}$, acting by the formula

Let $U$ be the kernel of the operator $P(\mathbf{T})$, and let $\mathbf{T}^{\prime }$ be the restriction of the operator $\mathbf{T}$ to $U$. Find

Prove that if the degree of the minimal polynomial of the operator А equals $k$, then there exists a vector $v$ such that the vectors $v, \mathbf{A} v, \ldots$..., $\mathrm{A}^{k-1} v$ are linearly independent.

Prove that the characteristic polynomial of an operator coincides with the minimal polynomial if and only if, in some basis, this operator has a matrix of the form

Find the characteristic polynomial of this operator.

Prove that the characteristic polynomial of the operator A coincides with its minimal polynomial if and only if every operator commuting with А can be represented in the form $P(\mathbf{A})$, where $P(x)$ is some polynomial.