For all $(M, N) \in (\mathcal{M}_n(\mathbb{R}))^2$, we denote $$(M \mid N) = \operatorname{tr}({}^t M N)$$ Prove that this defines an inner product on $\mathcal{M}_n(\mathbb{R})$. Explicitly express $(M \mid N)$ in terms of the coefficients of $M$ and $N$.

We denote by $\|M\|$ the Euclidean norm associated with the inner product $(M \mid N) = \operatorname{tr}({}^t M N)$. Fix $A \in \mathcal{M}_n(\mathbb{R})$, prove that there exists a matrix $M \in \mathcal{Y}_n$ such that: $$\forall N \in \mathcal{Y}_n \quad \|A - M\| \leqslant \|A - N\|$$

We denote by $\|M\|$ the Euclidean norm associated with the inner product $(M \mid N) = \operatorname{tr}({}^t M N)$. Justify the uniqueness of the matrix $M \in \mathcal{Y}_n$ minimizing $\|A - M\|$ over $\mathcal{Y}_n$ and explicitly express its coefficients in terms of those of $A$.

Justify that the determinant has a maximum on $\mathcal{X}_n$ (denoted $x_n$) and a maximum on $\mathcal{Y}_n$ (denoted $y_n$).

Prove that the sequence $(y_k)_{k \geqslant 2}$ is increasing, where $y_k$ denotes the maximum of the determinant on $\mathcal{Y}_k$.

Let $J \in \mathcal{X}_n$ be the matrix whose coefficients all equal 1. We set $M = J - I_n$.
Calculate $\operatorname{det}(M)$ and deduce that $\lim_{k \to +\infty} y_k = +\infty$.

Let $N = (n_{i,j})_{i,j} \in \mathcal{Y}_n$. Fix $1 \leqslant i, j \leqslant n$ and suppose that $n_{i,j} \in ]0,1[$.
Prove that by replacing $n_{i,j}$ either by 0 or by 1, we can obtain a matrix $N'$ of $\mathcal{Y}_n$ such that $\operatorname{det}(N) \leqslant \operatorname{det}(N')$.
Deduce that $x_n = y_n$.

Give two definitions of a vector isometry of $\mathbb{R}^n$ and prove their equivalence.

Prove that if $M \in \mathrm{O}_n(\mathbb{R})$, then its determinant equals 1 or $-1$. What do you think of the converse?

Prove that $\mathcal{P}_n = \mathcal{X}_n \cap \mathrm{O}_n(\mathbb{R})$ and determine its cardinality.

Let $\sigma$ and $\sigma'$ be two elements of $S_n$.
Prove that $P_\sigma P_{\sigma'} = P_{\sigma \circ \sigma'}$.
Justify that the application $\left\{\begin{array}{l} \mathbb{Z} \rightarrow S_n \\ k \mapsto \sigma^k \end{array}\right.$ is not injective.
Deduce that there exists an integer $N \geqslant 1$ such that $\sigma^N = \operatorname{Id}_{\{1,\ldots,n\}}$, where $\operatorname{Id}_{\{1,\ldots,n\}}$ denotes the identity map on the set $\{1,\ldots,n\}$.

For every matrix $A$ in $\mathcal{M}_n(\mathbb{K})$, we set $N(A) = \max_{1 \leqslant i \leqslant n}\left(\sum_{j=1}^{n}|a_{i,j}|\right)$. Show that the map $A \mapsto N(A)$ is a sub-multiplicative norm on $\mathcal{M}_n(\mathbb{K})$.

For every matrix $A$ in $\mathcal{M}_n(\mathbb{K})$, we set $N(A) = \max_{1 \leqslant i \leqslant n}\left(\sum_{j=1}^{n}|a_{i,j}|\right)$. Let $Q \in \mathrm{GL}_n(\mathbb{K})$. Show that $A \mapsto \|A\| = N\left(Q^{-1}AQ\right)$ is a sub-multiplicative norm on $\mathcal{M}_n(\mathbb{K})$.

We are given $A$ in $\mathcal{M}_n(\mathbb{C})$, with $\rho(A) < 1$. We want to show that $\lim_{m \rightarrow +\infty} A^m = 0$.
Let $P$ be in $\mathrm{GL}_n(\mathbb{C})$ and let $T$ be upper triangular, such that $A = PTP^{-1}$. We are given $\delta > 0$. We set $\Delta = \operatorname{diag}\left(1, \delta, \ldots, \delta^{n-1}\right)$ and $\widehat{T} = \Delta^{-1}T\Delta$.
Show that $\widehat{T}$ is upper triangular and that we can choose $\delta$ so that $N(\widehat{T}) < 1$.

We are given $A$ in $\mathcal{M}_n(\mathbb{C})$, with $\rho(A) < 1$. Let $P$ be in $\mathrm{GL}_n(\mathbb{C})$ and let $T$ be upper triangular, such that $A = PTP^{-1}$. We are given $\delta > 0$. We set $\Delta = \operatorname{diag}\left(1, \delta, \ldots, \delta^{n-1}\right)$ and $\widehat{T} = \Delta^{-1}T\Delta$.
With the choice of $\delta$ such that $N(\widehat{T}) < 1$, we set $Q = P\Delta$ and we equip $\mathcal{M}_n(\mathbb{C})$ with the norm $M \mapsto \|M\| = N\left(Q^{-1}MQ\right)$.
Show that $\|A\| < 1$ and deduce $\lim_{m \rightarrow +\infty} A^m = 0$.

Let $A$ denote a positive matrix in $\mathcal{M}_n(\mathbb{R})$.
Show that if there exists in $A$ a path from $i$ to $j$, with $i \neq j$, then there exists an elementary path from $i$ to $j$ of length $\ell \leqslant n-1$. Similarly, show that if there exists in $A$ a circuit passing through $i$, then there exists an elementary circuit passing through $i$ of length $\ell \leqslant n$.

Let $A \geqslant 0$ in $\mathcal{M}_n(\mathbb{R})$. Let $i, j$ be in $\llbracket 1, n \rrbracket$. Let $m \geqslant 1$. Show the equivalence of the propositions:

• there exists in $A$ a path with origin $i$, endpoint $j$, of length $m$;
• the entry with indices $i, j$ of $A^m$ (denoted $a_{i,j}^{(m)}$) is strictly positive.

You may proceed by induction on the integer $m \geqslant 1$.

Let $A$ denote a positive matrix in $\mathcal{M}_n(\mathbb{R})$. Let $i, j$ be in $\llbracket 1, n \rrbracket$, and let $\ell$ and $m$ be in $\mathbb{N}^*$. Show the equivalence of the propositions:

• there exists in $A^m$ a path with origin $i$, endpoint $j$, of length $\ell$;
• there exists in $A$ a path with origin $i$, endpoint $j$, of length $m\ell$.

Let $A$ be a primitive matrix in $\mathcal{M}_n(\mathbb{R})$.
Show that for all $i \neq j$ there exists in $A$ an elementary path from $i$ to $j$ of length $\ell \leqslant n-1$, and that for all $i$ there exists in $A$ an elementary circuit passing through $i$ of length $\ell \leqslant n$.

Give a simple example of a square matrix that is primitive but not strictly positive.

Let $B > 0$ in $\mathcal{M}_n(\mathbb{R})$ and $x \geqslant 0$ in $\mathbb{R}^n$ with $x \neq 0$. Show that $Bx > 0$.

Let $A$ be a primitive matrix and $m \in \mathbb{N}^*$ such that $A^m > 0$. Show that $\forall p \geqslant m, A^p > 0$. You may note, by justifying it, that none of the columns $c_1, c_2, \ldots, c_n$ of $A$ is zero.

Prove that if $A$ is primitive, then $A^k$ is primitive for all $k \geqslant 1$.

Show that the spectral radius of a primitive matrix is strictly positive.

We define the matrix $Z_n = (z_{i,j}) \in \mathcal{M}_n(\mathbb{R})$ by $z_{i,j} = \begin{cases} 1 & \text{if } 1 \leqslant i < n \text{ and } j = i+1 \\ 1 & \text{if } (i,j) \in \{(n-1,1),(n,2)\} \\ 0 & \text{in all other cases} \end{cases}$
Show that the characteristic polynomial of $Z_n$ is $X(X^{n-1} - 2)$.
Deduce that $Z_n$ is imprimitiv and specify its coefficient of imprimitivity.