Invariant subspaces and eigenvectors

Let $\left\{ A_{i} \mid 1 \leqslant i \leqslant m-1\right\}$ be an orthogonal system of idempotent matrices (see the remark at the end of item 1). Show that $A^{2}=A, A A_{i}=A_{i} A=A_{i}$, $1 \leqslant i \leqslant m-1$ for $A=A_{1}+A_{2}+\ldots +A_{m-1}$, and that if one sets $A_{m}=E-A$, then $\left\{ A_{i} \mid 1 \leqslant i \leqslant m\right\}$ is a complete orthogonal system.

Let $\mathcal{D}: M_{n}(\mathfrak {K}) \rightarrow M_{n}(\mathfrak {K})$ be a linear operator on the space of square matrices, not identically equal to $\mathcal{O}$, possessing the multiplicativity property:

for all $A, B \in M_{n}(\mathfrak {K})$. Prove that then $\mathcal{D}=f_{C}$ for some nonsingular $n \times n$ matrix $C$ (see Exercise 3 from § 1).

Let $\mathcal{A}: V \rightarrow V$ be a linear operator such that $\operatorname {Im} \mathcal{A}^{p}=\operatorname {Im} \mathcal{A}^{p+1}$ for some natural number $p$. Prove that in this case $V=\operatorname {Ker} \mathcal{A}^{p} \oplus \operatorname {Im} \mathcal{A}^{p}$ is a direct sum of two $\mathcal{A}$-invariant subspaces.

Prove that if the linear operators $\mathcal{E}, \mathcal{A}, \mathcal{A}^{2}, \ldots , \mathcal{A}^{n-1}$, acting on a vector space $V$ of dimension $n$, are linearly independent, then there exists a vector $\mathbf{v} \in V$ such that

(in this case one says that $V$ is cyclic).

Let $A$ be a real $n \times n$ matrix with no real eigenvalues, so that, in particular, $n$ is even and $A$ is invertible. Show that there exists a real matrix $B$ such that $A B=B A$ and $B^{2}=-E$, where $E$ is the identity matrix (D. Đoković).

Prove that for any $A, B \in M_{n}(\mathbb {R})$ the characteristic polynomials of the matrices $A B$ and $B A$ coincide.

Find the characteristic roots of the "cyclic" matrix

using the easily verified relation

Prove that the space $\operatorname {SMag}_{n}(\mathbb {Q})$ of semi-magic squares (see Example 8 from § 1, Ch. 1) is a $\mathbb {Q}$-subalgebra of $M_{n}(\mathbb {Q})$.

Let $A, B \in M_{n}(\mathfrak {K})$, $\operatorname {char} \mathfrak {K} \neq 2$. We write $B \sim A$ if $B=D A$, where $D=\operatorname {diag}\left(\theta_{1}, \ldots , \theta_{n}\right), \theta_{i}= \pm 1$. It is obvious that $\sim$ is an equivalence relation. If $S(A)$ is the equivalence class with representative $A$, then $\operatorname {Card} S(A) \leqslant 2^{n}$.

Prove that at least one of the matrices in $S(A)$ does not have the eigenvalue 1.

Let $\mathcal{A}$ be a linear operator on an $n$-dimensional vector space $V$. Show that