Deduce, using question 6, that if $p$ has a stable root then $J(p)$ is not invertible.

Show that $V A = B$ where $V \in \mathcal { M } _ { N } ( \mathbf { R } )$ is a matrix that you will make explicit.
Here $A = \left( a_1, a_2, \ldots, a_N \right)^\top \in \mathbf{R}^N$, $B = \left( \beta_0, \beta_1, \ldots, \beta_{N-1} \right)^\top \in \mathbf{R}^N$, and $\beta_k = \sum_{n=1}^{N} \lambda_n^k a_n$.

An example In this question, we further assume that $N = 4$ and $\ker(h - \operatorname{id}_V) = \{0\}$. For $j \in \{0, \ldots, 3\}$, we denote $V_j = \ker(h - \zeta^j \operatorname{id}_V)$. According to $11^\circ$b), the data of $u$ is equivalent to the data of the two linear maps $u_1 : V_1 \rightarrow V_2$ and $u_2 : V_2 \rightarrow V_3$ induced by $u$.
a) Verify that $u^3 = 0$.
b) 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.
c) Prove that the number of blocks of each type is determined by the data of the three dimensions $d_j = \dim V_j$ ($1 \leqslant j \leqslant 3$) and the three ranks $r_1 = \operatorname{rg} u_1$, $r_2 = \operatorname{rg} u_2$ and $r_{21} = \operatorname{rg}(u_2 \circ u_1)$.

Let $\left(\mathbf{v}_1, \ldots, \mathbf{v}_n\right)$ be any orthonormal basis of $\mathbb{R}^n$. Show that $$\mathbb{I}_n = \sum_{k=1}^n \mathbf{v}_k \mathbf{v}_k^T$$

Let $(u,v,r,s) \in (\mathbb{N}^*)^4$. Let $A = (a_{ij})_{\substack{1 \leqslant i \leqslant u \\ 1 \leqslant j \leqslant v}} \in \mathcal{M}_{u,v}(\mathbb{R})$ and $B \in \mathcal{M}_{r,s}(\mathbb{R})$. We define the Kronecker product of $A$ by $B$, and we denote $A \otimes B$, the matrix of $\mathcal{M}_{ur,vs}(\mathbb{R})$ which is defined by $uv$ blocks of size $r \times s$ in such a way that, for all $(i,j) \in \llbracket 1,u \rrbracket \times \llbracket 1,v \rrbracket$, the block with index $(i,j)$ is $a_{i,j}B$.
Besides $u,v,r$ and $s$, we are also given two non-zero natural integers, $w$ and $t$.
Show that, for all matrices $A \in \mathcal{M}_{u,v}(\mathbb{R})$, $A' \in \mathcal{M}_{v,w}(\mathbb{R})$, $B \in \mathcal{M}_{r,s}(\mathbb{R})$ and $B' \in \mathcal{M}_{s,t}(\mathbb{R})$, $$(A \otimes B)(A' \otimes B') = (AA') \otimes (BB').$$

Let $\left(\mathbf{v}_1, \ldots, \mathbf{v}_n\right)$ be any orthonormal basis of $\mathbb{R}^n$. Show that $$\mathbb{I}_n = \sum_{k=1}^n \mathbf{v}_k \mathbf{v}_k^T.$$

Let $n \geq 1$ be an integer and $P \subset \mathbb{R}^n$ a polytope. Let $V$ be the set of vertices of $P$. Show that $P = \operatorname{Conv}(V)$.

In this question, we further assume that $N = 4$ and $\ker(h - \mathrm{id}_V) = \{0\}$. Verify that $u^3 = 0$.

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.

Prove that the number of blocks of each type is determined by the data of the three dimensions $d_j = \dim V_j$ ($1 \leqslant j \leqslant 3$) and the three ranks $r_1 = \operatorname{rg} u_1$, $r_2 = \operatorname{rg} u_2$ and $r_{21} = \operatorname{rg}(u_2 \circ u_1)$.

For every $\lambda \in \mathbb{C}$ nonzero, we set $J_n(\lambda) = \lambda I_n + N$, where $N = (n_{i,j})_{1 \leq i,j \leq n}$ with $n_{i,j} = 1$ if $j = i+1$ and $n_{i,j} = 0$ otherwise.
Prove that $J_n(\lambda)$ is invertible and determine in terms of $N$ and $\lambda$ the matrix $N'$ such that $J_n(\lambda)^{-1} = \frac{1}{\lambda} I_n + N'$.

For every $\lambda \in \mathbf{C}$ nonzero, we set $J_n(\lambda) = \lambda I_n + N$. Prove that $J_n(\lambda)$ is invertible and determine in terms of $N$ and $\lambda$ the matrix $N'$ such that $J_n(\lambda)^{-1} = \frac{1}{\lambda} I_n + N'$.

Let two matrices $A, B \in \mathcal{M}_n(\mathbf{R})$ such that there exists a matrix $P \in GL_n(\mathbf{R})$ satisfying $A = P^\top B P$. Show that $d(B) \geq d(A)$ then that $d(B) = d(A)$.

Prove that the system $V A = B$ admits a unique solution $A \in \mathbf { R } ^ { N }$.
Here $V$ is the matrix from question 14, with entries $V_{k,n} = \lambda_n^{k-1}$ for $k = 1,\ldots,N$ and $n = 1,\ldots,N$, and the $\lambda_n$ are strictly increasing real numbers.

A reduction We fix two nonzero natural integers $m$ and $n$. For $(A, B)$ in $\mathcal{M}_{n,m}(\mathbb{C}) \times \mathcal{M}_{m,n}(\mathbb{C})$ we define the following $(m+n) \times (m+n)$ matrices: $$M_{A,B} = \begin{pmatrix} 0_m & B \\ A & 0_n \end{pmatrix} \quad \text{and} \quad H = \begin{pmatrix} \mathrm{I}_m & 0_{m,n} \\ 0_{n,m} & -\mathrm{I}_n \end{pmatrix}.$$ 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 $H = RHR^{-1}$.

We now consider the symmetric matrix $A$. By virtue of the spectral theorem, we denote by $\lambda_1 \leqslant \cdots \leqslant \lambda_n$ the eigenvalues of $A$, and $\left(\mathbf{w}_1, \ldots, \mathbf{w}_n\right)$ a corresponding orthonormal basis of eigenvectors.
(a) Show that $$A = \sum_{k=1}^n \lambda_k \mathbf{w}_k \mathbf{w}_k^T$$ (b) Show that for all $x \in \mathbb{R} \backslash \left\{\lambda_1, \ldots, \lambda_n\right\}$, we have $$\left(x \mathbb{I}_n - A\right)^{-1} = \sum_{k=1}^n \frac{1}{x - \lambda_k} \mathbf{w}_k \mathbf{w}_k^T$$

Let $(u,v,r,s) \in (\mathbb{N}^*)^4$. Let $A = (a_{ij})_{\substack{1 \leqslant i \leqslant u \\ 1 \leqslant j \leqslant v}} \in \mathcal{M}_{u,v}(\mathbb{R})$ and $B \in \mathcal{M}_{r,s}(\mathbb{R})$. We define the Kronecker product of $A$ by $B$, and we denote $A \otimes B$, the matrix of $\mathcal{M}_{ur,vs}(\mathbb{R})$ which is defined by $uv$ blocks of size $r \times s$ in such a way that, for all $(i,j) \in \llbracket 1,u \rrbracket \times \llbracket 1,v \rrbracket$, the block with index $(i,j)$ is $a_{i,j}B$.
Show that if $A \in \mathcal{M}_u(\mathbb{R})$ and $B \in \mathcal{M}_r(\mathbb{R})$ are diagonalizable, then $A \otimes B$ is diagonalizable and $$\operatorname{Sp}(A \otimes B) = \{\lambda\mu \mid \lambda \in \operatorname{Sp}(A), \mu \in \operatorname{Sp}(B)\}.$$ One may start by determining the inverse of the Kronecker product of two invertible matrices.

We now focus on the symmetric matrix $A \in \mathcal{S}_n(\mathbb{R})$. By virtue of the spectral theorem, we denote by $\lambda_1 \leqslant \cdots \leqslant \lambda_n$ the eigenvalues of $A$, and $\left(\mathbf{w}_1, \ldots, \mathbf{w}_n\right)$ a corresponding orthonormal basis of eigenvectors.
(a) Show that $$A = \sum_{k=1}^n \lambda_k \mathbf{w}_k \mathbf{w}_k^T.$$
(b) Show that for all $x \in \mathbb{R} \backslash \left\{\lambda_1, \ldots, \lambda_n\right\}$, we have $$\left(x \mathbb{I}_n - A\right)^{-1} = \sum_{k=1}^n \frac{1}{x - \lambda_k} \mathbf{w}_k \mathbf{w}_k^T.$$

Let $V \subset \mathbb{R}^n$ be a non-empty finite set. Justify that to prove that $\operatorname{Conv}(V)$ is a polytope it suffices to treat the case where $\operatorname{Conv}(V)$ is not contained in a hyperplane of $\mathbb{R}^n$ and contains 0 in its interior.

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 $H = RHR^{-1}$.

For every $\lambda \in \mathbb{C}$ nonzero, we set $J_n(\lambda) = \lambda I_n + N$, where $N = (n_{i,j})_{1 \leq i,j \leq n}$ with $n_{i,j} = 1$ if $j = i+1$ and $n_{i,j} = 0$ otherwise, and $N'$ is the matrix such that $J_n(\lambda)^{-1} = \frac{1}{\lambda} I_n + N'$.
Calculate $(N')^n$ and deduce that $J_n(\lambda)^{-1}$ is similar to $J_n\left(\frac{1}{\lambda}\right)$.

For every $\lambda \in \mathbf{C}$ nonzero, we set $J_n(\lambda) = \lambda I_n + N$, and $N'$ is the matrix such that $J_n(\lambda)^{-1} = \frac{1}{\lambda} I_n + N'$. Calculate $(N')^n$ and deduce that $J_n(\lambda)^{-1}$ is similar to $J_n\left(\frac{1}{\lambda}\right)$.

For every matrix $M \in S_n(\mathbf{R})$ construct a vector subspace $F_M$ of $\mathcal{M}_{n,1}(\mathbf{R})$ of dimension $\pi(M)$ satisfying condition $(\mathcal{C}_M)$: $$\forall X \in F \setminus \{0_{n,1}\} \quad X^\top M X > 0.$$ We thus have $d(M) \geq \pi(M)$.

Two linear maps: decomposition We fix two nonzero natural integers $m$ and $n$, matrices $(A,B) \in \mathcal{M}_{n,m}(\mathbb{C}) \times \mathcal{M}_{m,n}(\mathbb{C})$, and denote $M = M_{A,B} = \begin{pmatrix} 0_m & B \\ A & 0_n \end{pmatrix}$ and $H = \begin{pmatrix} \mathrm{I}_m & 0_{m,n} \\ 0_{n,m} & -\mathrm{I}_n \end{pmatrix}$.
a) Calculate $H^2$, $HMH^{-1}$ and, for a polynomial $P$ of $\mathbb{C}[X]$, calculate $HP(M)H^{-1}$.
b) Prove that if a complex number $\lambda$ is an eigenvalue of $M$, then $-\lambda$ is also an eigenvalue of $M$ with the same multiplicity.
c) Let $\chi_M$ be the characteristic polynomial of $M$. We write it as $\chi_M = X^r Q$ where $r$ is an integer and $Q$ is a polynomial whose constant coefficient is nonzero. Briefly justify that $$\mathbb{C}^{m+n} = \ker M^r \oplus \ker Q(M)$$ and verify that these subspaces are stable under $H$.

Let $\lambda$ be an eigenvalue of $A$ with multiplicity $m \geqslant 2$. We set $E = \operatorname{Ker}\left(A - \lambda \mathbb{I}_n\right)$.
(a) Show that $\operatorname{dim}\left(E \cap \{\mathbf{u}\}^\perp\right) \geqslant m - 1$.
(b) Deduce that $\lambda$ is an eigenvalue of $B$ with multiplicity at least $m - 1$.