We now consider the case where $A \in \mathcal{S}_n(\mathbb{R})$ is symmetric. Let $\mathbf{u} \in \mathbb{R}^n$ be such that $\|\mathbf{u}\| = 1$. We set $B = A + \mathbf{u}\mathbf{u}^T$. Show that $B \in \mathcal{S}_n(\mathbb{R})$.

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 $\otimes$ is a bilinear map from $\mathcal{M}_{u,v}(\mathbb{R}) \times \mathcal{M}_{r,s}(\mathbb{R})$ to $\mathcal{M}_{ur,vs}(\mathbb{R})$.

We consider the case where $A \in \mathcal{S}_n(\mathbb{R})$ is symmetric. Let $\mathbf{u} \in \mathbb{R}^n$ be such that $\|\mathbf{u}\| = 1$. We set $B = A + \mathbf{u u}^T$. Show that $B \in \mathcal{S}_n(\mathbb{R})$.

Let $E$ be a $\mathbb{C}$-vector space of dimension $n$. Let $g$ be an endomorphism of $E$ such that $g^n = 0$ and $g^{n-1} \neq 0$.
Prove that there exists a basis of $E$ in which the matrix of $g$ is the matrix $N$ below: $$N = \left(\begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ \vdots & & & \ddots & 1 \\ 0 & \ldots & & \ldots & 0 \end{array}\right)$$ In other words: $N = (n_{i,j})_{1 \leq i,j \leq n}$ with $n_{i,j} = 1$ if $j = i+1$ and $n_{i,j} = 0$ otherwise.

Let $E$ be a $\mathbf{C}$-vector space of dimension $n$. Let $g$ be an endomorphism of $E$ such that $g^n = 0$ and $g^{n-1} \neq 0$. Prove that there exists a basis of $E$ in which the matrix of $g$ is the matrix $N$ below: $$N = \left(\begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ \vdots & & & \ddots & 1 \\ 0 & \ldots & & \ldots & 0 \end{array}\right)$$ In other words: $N = (n_{i,j})_{1 \leq i,j \leq n}$ with $n_{i,j} = 1$ if $j = i+1$ and $n_{i,j} = 0$ otherwise.

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 $\beta_k = \sum_{n=1}^{N} \lambda_n^k a_n$.

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

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 $$\left(x \mathbb{I}_n - A\right)^{-1} = \sum_{k=1}^n \frac{1}{x - \lambda_k} \mathbf{w}_k \mathbf{w}_k^T.$$

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 $m - 1$.

We set $N = n^2$ and $$J_N^{(2)} = I_n \otimes J_n^{(1)} + J_n^{(1)} \otimes I_n \in \mathcal{M}_N(\mathbb{R})$$
Verify that, in the case where $n = 3$, $J_N^{(2)}$ is the matrix such that, for all $(i,j) \in \llbracket 1,9 \rrbracket^2$, the coefficient with index $(i,j)$ equals 1 if the vertices $i$ and $j$ of the graph are connected by an edge (the edges count whether they are dashed or solid), and equals 0 otherwise.

We consider $A \in \mathcal{S}_n(\mathbb{R})$ symmetric with $B = A + \mathbf{u u}^T$ where $\|\mathbf{u}\| = 1$. 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 $m - 1$.

Calculate $H^2$, $HMH^{-1}$ and, for a polynomial $P$ of $\mathbb{C}[X]$, calculate $HP(M)H^{-1}$.

For every polynomial $P = P(X) \in \mathbf{C}_{n-1}[X]$ we set $$\left\{\begin{array}{l} s_1(P) = P(-X) \\ s_2(P) = P(1-X) \\ g(P) = P(X+1) - P(X) \end{array}\right.$$ We thus define three endomorphisms of the vector space $\mathbb{C}_{n-1}[X]$.
Calculate $s_1^2$, $s_2^2$ and express $s_1 \circ s_2$ in terms of $g$ and $Id_{\mathbb{C}_{n-1}[X]}$.