Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let $\varepsilon_0 > 0$ be such that for all $t \in ]-\varepsilon_0, \varepsilon_0[, A + tM \in S_n^{++}(\mathrm{R})$. Let $\alpha \in ]-\frac{1}{n}, +\infty[\backslash\{0\}$. We define the application $\varphi_\alpha$ by $$\forall t \in ]-\varepsilon_0, \varepsilon_0[, \quad \varphi_\alpha(t) = \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM).$$ Show that $\varphi_\alpha$ is differentiable on $]-\varepsilon_0, \varepsilon_0[$ and that $$\forall t \in ]-\varepsilon_0, \varepsilon_0[, \quad \varphi_\alpha'(t) = -\operatorname{Tr}\left((A + tM)^{-1}M\right) \operatorname{det}^{-\alpha}(A + tM).$$

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$. Let the application $f _ { A }$ defined on $\mathbf { R }$ by
$$f _ { A } ( t ) = \operatorname { det } ( A + t M )$$
and let $\varepsilon_0 > 0$ be such that for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [ , A + t M \in S _ { n } ^ { + + } ( \mathbf { R } )$. Determine $f _ { A } ^ { \prime } ( t )$ for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [$.

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$, and let $\varepsilon_0 > 0$ be such that for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [ , A + t M \in S _ { n } ^ { + + } ( \mathbf { R } )$. Let $\alpha \in ] - \frac { 1 } { n } , + \infty \left[ \backslash \{ 0 \} \right.$. We define the application $\varphi _ { \alpha }$ by
$$\forall t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } \left[ , \varphi _ { \alpha } ( t ) = \frac { 1 } { \alpha } \operatorname { det } ^ { - \alpha } ( A + t M ) \right.$$
Show that $\varphi _ { \alpha }$ is differentiable on $] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [$ and that
$$\forall t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } \left[ , \quad \varphi _ { \alpha } ^ { \prime } ( t ) = - \operatorname { Tr } \left( ( A + t M ) ^ { - 1 } M \right) \operatorname { det } ^ { - \alpha } ( A + t M ) . \right.$$

Let $J \in M_{n}(\mathbb{C})$ be the matrix defined by
$$J = \left( \begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & 0 & 1 \\ 1 & 0 & \ldots & 0 & 0 \end{array} \right).$$
Deduce the value of the determinant
$$\left| \begin{array}{ccccc} a_{0} & a_{1} & \ldots & a_{n-2} & a_{n-1} \\ a_{n-1} & a_{0} & \ddots & & a_{n-2} \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ a_{2} & & \ddots & a_{0} & a_{1} \\ a_{1} & a_{2} & \cdots & a_{n-1} & a_{0} \end{array} \right|$$
where $a_{0}, \ldots, a_{n-1}$ are arbitrary complex numbers.

Show that an adjacency matrix of a non-empty graph is never of rank 1.

Show that an adjacency matrix of a graph whose non-isolated vertices form a star-type graph is of rank 2 and represent an example of a graph whose adjacency matrix is of rank 2 and which is not of the previous type.

Let $n$ be a natural integer with $n \geqslant 2$. For any real number $x$, we consider the following matrix in $\mathscr{M}_{n}(\mathbb{R})$ $$M_{x} = \left(\begin{array}{ccccc} x & 1 & \cdots & 1 & 1 \\ 1 & x & \cdots & 1 & 1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \cdots & x & 1 \\ 1 & 1 & \cdots & 1 & x \end{array}\right)$$ Deduce that for all $x \in \mathbb{R}$, we have $$\sum_{\sigma \in \mathfrak{S}_{n}} \varepsilon(\sigma) x^{\nu(\sigma)} = (x-1)^{n-1}(x+n-1)$$

Let $n$ be a natural integer with $n \geqslant 2$. For any real number $x$, we consider the following matrix in $\mathscr{M}_n(\mathbb{R})$ $$M_x = \left(\begin{array}{ccccc} x & 1 & \cdots & 1 & 1 \\ 1 & x & \cdots & 1 & 1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \cdots & x & 1 \\ 1 & 1 & \cdots & 1 & x \end{array}\right).$$ Deduce that for all $x \in \mathbb{R}$, we have $$\sum_{\sigma \in \mathfrak{S}_n} \varepsilon(\sigma) x^{\nu(\sigma)} = (x-1)^{n-1}(x+n-1).$$

Let $R \in \mathrm{O}_{d}(\mathbb{R})$. Verify that $\operatorname{det}(R) \in \{-1, +1\}$.

Let $A$ be a matrix belonging to $\mathbf{GL}_n$. Let $x$ be a nonzero real number. Express $\det(x I_n - A)$ in terms of $x$, $\det A$ and $\det\left(\frac{1}{x} I_n - A^{-1}\right)$.

Let $A$ be a matrix belonging to $\mathbf{GL}_n$. Let $x$ be a nonzero real number. Express $\det(x I_n - A)$ in terms of $x$, $\det A$ and $\det\left(\frac{1}{x} I_n - A^{-1}\right)$.

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

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 $d(B) = d(A)$.

Deduce that the matrix $J(p)$ is invertible if and only if $p$ has no stable root.

We assume in this part that all roots of $p$ are stable and have multiplicity 1 and we denote by $h = Xp'$ (where $p'$ is the derivative polynomial of $p$) and $h_0$ the reciprocal polynomial of $h$. We recall that, according to question 3, there exists a real number $\lambda \in \{-1, 1\}$ such that $p = \lambda p_0$.
Show that $J(h)$ is invertible.

Let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n \backslash \{\mathbf{0}\}$. We set $M = \mathbf{u v}^T$. Show that $M$ is a square matrix of size $n \times n$, of rank 1.

Calculate with justification the rank of the following matrix $J \in \mathcal{M}_n(\mathbb{R})$: $$J = \left(\begin{array}{cccc} 1 & 1 & \cdots & 1 \\ 1 & 1 & \cdots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \cdots & 1 \end{array}\right)$$

Conversely, let $K \in \mathcal{M}_n(\mathbb{R})$ be a square matrix of rank 1. Show that there exist $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n \backslash \{\mathbf{0}\}$ such that $K = \mathbf{u v}^T$.

Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Show that $$\operatorname{det}\left(\mathbb{I}_n + \mathbf{u}\mathbf{v}^T\right) = 1 + \langle \mathbf{v}, \mathbf{u} \rangle$$

Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Show more generally that $$\operatorname{det}\left(A + \mathbf{u}\mathbf{v}^T\right) = \operatorname{det}(A)\left(1 + \left\langle \mathbf{v}, A^{-1}\mathbf{u} \right\rangle\right).$$

Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Let $C \in \mathcal{M}_n(\mathbb{R})$ be a matrix such that $\operatorname{det}(C) = 0$. Is it always true that $\operatorname{det}\left(C + \mathbf{u v}^T\right) = 0$? Justify your answer.

We denote by $\chi_A(x) = \operatorname{det}\left(x \mathbb{I}_n - A\right)$ the characteristic polynomial of $A$, and $\chi_B(x) = \operatorname{det}\left(x \mathbb{I}_n - B\right)$ that of $B$. Show that, for all $x \in \mathbb{R} \backslash \left\{\lambda_1, \ldots, \lambda_n\right\}$, we have $$\chi_B(x) = \chi_A(x)\left(1 - \sum_{k=1}^n \frac{\left\langle \mathbf{w}_k, \mathbf{u} \right\rangle^2}{x - \lambda_k}\right).$$

Let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n \backslash \{\mathbf{0}\}$. We set $M = \mathbf{u v}^T$. Show that $M$ is a square matrix of size $n \times n$, of rank 1.

Calculate with justification the rank of the following matrix $J \in \mathcal{M}_n(\mathbb{R})$: $$J = \left(\begin{array}{cccc} 1 & 1 & \cdots & 1 \\ 1 & 1 & \cdots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \cdots & 1 \end{array}\right).$$

Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Show that $$\operatorname{det}\left(\mathbb{I}_n + \mathbf{u v}^T\right) = 1 + \langle \mathbf{v}, \mathbf{u} \rangle.$$