Prove that $h$ is diagonalizable.

Prove that there exists a basis of $V$ in which the matrices of $u$ and $h$ are block diagonal and the diagonal blocks are respectively of the form $$J_r \quad \text{and} \quad D_{r,a} = \operatorname{diag}(\zeta^a, \zeta^{a+1}, \ldots, \zeta^{a+r-1})$$ for $r \in \mathbb{N}^*$ and $a \in \{0, \ldots, N-1\}$ suitable.

Construct pairs $(u, h)$ that give rise to six different types of pairs of diagonal blocks $(J_r, D_{r,a})$ in the graded version of the decomposition theorem.

Let $(A, B)$ and $(A', B')$ be two pairs of matrices in $\mathcal{M}_{n,m}(\mathbb{C}) \times \mathcal{M}_{m,n}(\mathbb{C})$. Prove that the following conditions are equivalent: (i) $(A, B)$ and $(A', B')$ are simultaneously equivalent; (ii) there exist $P \in \mathrm{GL}_m(\mathbb{C})$ and $Q \in \mathrm{GL}_n(\mathbb{C})$ such that $A' = QAP^{-1}$ and $B' = PBQ^{-1}$; (iii) there exists $R \in \mathrm{GL}_{m+n}(\mathbb{C})$ such that $M_{A',B'} = RM_{A,B}R^{-1}$ and $H = RHR^{-1}$.

In this question, we assume that $M$ is nilpotent. Prove that $(M, H)$ is simultaneously similar to a pair of block diagonal matrices whose diagonal blocks are respectively of the form $$\left(\begin{array}{cc} 0_r & B_0 \\ A_0 & 0_s \end{array}\right) \quad \text{and} \quad \left(\begin{array}{cc} \mathrm{I}_r & 0 \\ 0 & -\mathrm{I}_s \end{array}\right),$$ where $r$ and $s$ are natural integers with $|r - s| \leqslant 1$ and $A_0$ and $B_0$ form one of the following pairs: $$A_0 = \left(\begin{array}{ccccc} 1 & 0 & \cdots & 0 & 0 \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{array}\right)_{s \times (s+1)} \quad \text{and} \quad B_0 = \left(\begin{array}{ccccc} 0 & \cdots & \cdots & 0 \\ 1 & \ddots & & \vdots \\ 0 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & 1 \end{array}\right)_{(s+1) \times s};$$ $$A_0 = \mathrm{I}_r \quad \text{and} \quad B_0 = J_r;$$ $$A_0 = J_r \quad \text{and} \quad B_0 = \mathrm{I}_r;$$ $$A_0 = \left(\begin{array}{cccc} 0 & \cdots & \cdots & 0 \\ 1 & \ddots & & \vdots \\ 0 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & 1 \end{array}\right)_{(r+1) \times r} \quad \text{and} \quad B_0 = \left(\begin{array}{ccccc} 1 & 0 & \cdots & 0 & 0 \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{array}\right)_{r \times (r+1)}.$$

Prove that $(M, H)$ is simultaneously similar to a pair of block diagonal matrices whose diagonal blocks are of even size and are respectively of the form $$\left(\begin{array}{cc} 0_r & B_1 \\ A_1 & 0_r \end{array}\right) \quad \text{and} \quad \left(\begin{array}{cc} \mathrm{I}_r & 0_r \\ 0_r & -\mathrm{I}_r \end{array}\right),$$ where $$A_1 = \mathrm{I}_r \quad \text{and} \quad B_1 = \lambda \mathrm{I}_r + J_r$$ for $r$ nonzero integer and $\lambda$ nonzero complex suitable.

Which of the following is(are) NOT the square of a $3 \times 3$ matrix with real entries?
[A] $\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
[B] $\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{array}\right]$
[C] $\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{array}\right]$
[D] $\left[\begin{array}{ccc}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{array}\right]$

Define a relation $R$ over a class of $n \times n$ real matrices $A$ and $B$ as ``$ARB$ iff there exists a non-singular matrix $P$ such that $PAP^{-1} = B$''. Then which of the following is true?