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 $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [$.

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$ and we set $\chi = \operatorname { det } \left( X I _ { n } - M \right) \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$.
Show that $M _ { \mid t = 0 }$ admits a real eigenvalue.

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let $\varepsilon_0 > 0$ be such that for all $t \in ]-\varepsilon_0, \varepsilon_0[, A + tM \in S_n^{++}(\mathrm{R})$. We admit that the function $\Phi : t \mapsto (A + tM)^{-1}$ is of class $C^1$ on $]-\varepsilon_0, \varepsilon_0[$. By noting that $\Phi(t) \times (A + tM) = I_n$, show that $$\Phi(t) \underset{t \rightarrow 0}{=} A^{-1} - A^{-1}MA^{-1} t + o(t).$$

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$, and let $\varepsilon_0 > 0$ be such that for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [ , A + t M \in S _ { n } ^ { + + } ( \mathbf { R } )$. We admit that the function $\Phi : t \mapsto ( A + t M ) ^ { - 1 }$ is of class $C ^ { 1 }$ on $] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [$. By noting that $\Phi ( t ) \times ( A + t M ) = I _ { n }$, show that
$$\Phi ( t ) \underset { t \rightarrow 0 } { = } A ^ { - 1 } - A ^ { - 1 } M A ^ { - 1 } t + o ( t )$$

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$ and we set $\chi = \operatorname { det } \left( X I _ { n } - M \right) \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$. By Theorem 1, there exists $\rho _ { 1 } \in \mathbb { R } _ { + } ^ { * }$, $\rho _ { 1 } \leqslant \rho$ such that $\chi$ factors in the form $\chi = F G$ with $F \in \mathscr { D } _ { \rho _ { 1 } } \left( \mathbb { R } _ { d } [ X ] \right)$ and $G \in \mathscr { D } _ { \rho _ { 1 } } \left( \mathbb { R } _ { n - d } [ X ] \right)$ and $F _ { \mid t = 0 } = ( X - \lambda ) ^ { d }$.
Only in this question, we assume that $d = n$. Show that there exists a symmetric matrix $M _ { 0 } \in \mathscr { D } _ { \rho _ { 1 } } \left( S _ { n } ( \mathbb { R } ) \right)$ such that $M = \lambda I _ { n } + t M _ { 0 }$ for all $t \in U _ { \rho _ { 1 } }$.

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 $M \in S _ { n } ( \mathbf { R } )$, 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
$$\forall t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } \left[ , \quad \varphi _ { \alpha } ^ { \prime } ( t ) = - \operatorname { Tr } \left( ( A + t M ) ^ { - 1 } M \right) \operatorname { det } ^ { - \alpha } ( A + t M ) . \right.$$

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$ and $B = G ( M )$; we thus have $A , B \in \mathscr { D } _ { \rho _ { 1 } } \left( S _ { n } ( \mathbb { R } ) \right)$. For $a \in U _ { \rho _ { 1 } }$, we set $A _ { a } = A _ { \mid t = a }$ and $B _ { a } = B _ { \mid t = a }$.
Show that there exist two matrices $U \in \mathscr { M } _ { n , d } ( \mathbb { R } )$ and $V \in \mathscr { M } _ { n , n - d } ( \mathbb { R } )$ such that:

• $\operatorname { im } \left( B _ { 0 } U \right) = \operatorname { im } \left( B _ { 0 } \right)$,
• $\operatorname { im } \left( A _ { 0 } V \right) = \operatorname { im } \left( A _ { 0 } \right)$ and
• the block matrix $\left( B _ { 0 } U \mid A _ { 0 } V \right)$ is invertible.

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$ and $B = G ( M )$, and $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 1 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$ where $U \in \mathscr { M } _ { n , d } ( \mathbb { R } )$ and $V \in \mathscr { M } _ { n , n - d } ( \mathbb { R } )$ are as in question 20.
Show that there exists $\rho _ { 2 } \in \mathbb { R } _ { + } ^ { * }$, $\rho _ { 2 } \leqslant \rho _ { 1 }$ such that $Q \in \operatorname { GL } _ { n } \left( \mathscr { D } _ { \rho _ { 2 } } ( \mathbb { R } ) \right)$. (One may use the result of question 6.)

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Show that $A^{-1}M$ is similar to a real symmetric matrix.
Hint: You may use question 3.

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$. Show that $A ^ { - 1 } M$ is similar to a real symmetric matrix.
Hint: You may use question 3.

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$, $B = G ( M )$, $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$. For $a \in U _ { \rho _ { 2 } }$, we set $A _ { a } = A _ { \mid t = a }$ and $B _ { a } = B _ { \mid t = a }$. We consider a real number $a \in U _ { \rho _ { 2 } }$.
22a. Show that $\operatorname { im } \left( B _ { a } U \right) \oplus \operatorname { im } \left( A _ { a } V \right) = \mathbb { R } ^ { n }$.
22b. Show the equalities:

• $\operatorname { im } \left( B _ { a } U \right) = \operatorname { im } \left( B _ { a } \right) = \operatorname { ker } \left( A _ { a } \right)$ and
• $\operatorname { im } \left( A _ { a } V \right) = \operatorname { im } \left( A _ { a } \right) = \operatorname { ker } \left( B _ { a } \right)$.

(One may begin by showing the inclusions from left to right, then use a dimension argument.)

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let $\alpha \in ]-\frac{1}{n}, +\infty[\backslash\{0\}$ and $\varphi_\alpha(t) = \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM)$. Using the fact that $A^{-1}M$ is similar to a real symmetric matrix, deduce that $\varphi_\alpha''(0) \geq 0$.

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$, and let $\alpha \in ] - \frac { 1 } { n } , + \infty \left[ \backslash \{ 0 \} \right.$. With $\varphi _ { \alpha } ^ { \prime \prime } ( 0 ) = \operatorname { det } ^ { - \alpha } ( A ) \left( \alpha \operatorname { Tr } ^ { 2 } \left( A ^ { - 1 } M \right) + \operatorname { Tr } \left( \left( A ^ { - 1 } M \right) ^ { 2 } \right) \right)$, deduce that $\varphi _ { \alpha } ^ { \prime \prime } ( 0 ) \geq 0$.

We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$, $B = G ( M )$, $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$.
Show that $Q ^ { - 1 } \cdot M \cdot Q = \operatorname { Diag } \left( M _ { 1 } , M _ { 2 } \right)$ with $M _ { 1 } \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { d } ( \mathbb { R } ) \right) , M _ { 2 } \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n - d } ( \mathbb { R } ) \right)$.

Let $Q$ be a delta endomorphism, let $(q_n)_{n \in \mathbb{N}}$ be the sequence of polynomials associated with $Q$, and let $n$ be a natural number. The family $(q_0, q_1, \ldots, q_n)$ is a basis of $\mathbb{K}_n[X]$.
According to question 23, $Q$ induces an endomorphism of $\mathbb{K}_n[X]$ denoted $Q_n$. Give its matrix in the previous basis. Deduce its trace, its determinant and its characteristic polynomial.

Give examples of Hadamard matrices of order 1 and 2.

Show that the matrices $M$ and $\left( m _ { \rho ( i ) , \rho ( j ) } \right) _ { 1 \leq i , j \leq n }$ are similar. Deduce that if $G = ( S , A )$ is a non-empty graph, and if $\sigma$ and $\sigma ^ { \prime }$ are two indexings of $S$, then $M _ { G , \sigma }$ and $M _ { G , \sigma ^ { \prime } }$ are similar.

Give a necessary and sufficient condition on $R_u$ for $\mathbb{M}_n(u) = \emptyset$ and give an example of $u$ for which this equality holds.

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 $\operatorname{det}(R) \in \{-1, +1\}$.

Show that if $H$ is a Hadamard matrix then any matrix obtained by multiplying a row or column of $H$ by $-1$ or by exchanging two rows or two columns of $H$ is still a Hadamard matrix.

Show that $\mathbb{M}_n(u) \neq \{0_n\}$.

Verify that $(A, B) \mapsto \langle A, B \rangle$ is an inner product on the vector space $\mathscr{M}_{d}(\mathbb{R})$. We denote by $\|A\| = \sqrt{\langle A, A \rangle}$ the associated norm.