Let $A = \left( a _ { i j } \right) _ { 1 \leqslant i , j \leqslant n } \in \mathcal { M } _ { n } ( \mathbb { R } )$. We define $R ( A ) = \left\{ { } ^ { t } X A X \mid X \in \mathbb { R } ^ { n } , \| X \| = 1 \right\}$.
We consider two real numbers $a \in R ( A )$ and $b \in R ( A )$, with $a < b$. Let $X _ { 1 }$ and $X _ { 2 }$ be two vectors of norm 1 such that ${ } ^ { t } X _ { 1 } A X _ { 1 } = a$, ${ } ^ { t } X _ { 2 } A X _ { 2 } = b$.
I.C.1) Prove that $X _ { 1 }$ and $X _ { 2 }$ are linearly independent.
I.C.2) We set $X _ { \lambda } = \lambda X _ { 1 } + ( 1 - \lambda ) X _ { 2 }$ for $0 \leqslant \lambda \leqslant 1$.
Prove that the function $\phi : \lambda \mapsto \frac { { } ^ { t } X _ { \lambda } A X _ { \lambda } } { \left\| X _ { \lambda } \right\| ^ { 2 } }$ is defined and continuous on the interval $[ 0,1 ]$.
I.C.3) Deduce that the segment $[ a , b ]$ is included in $R ( A )$.

Let $A = \left( a _ { i j } \right) _ { 1 \leqslant i , j \leqslant n } \in \mathcal { M } _ { n } ( \mathbb { R } )$. We define $R ( A ) = \left\{ { } ^ { t } X A X \mid X \in \mathbb { R } ^ { n } , \| X \| = 1 \right\}$.
Prove that if $\operatorname { Tr } ( A ) = 0$ then $0 \in R ( A )$.

Let $A = \left( a _ { i j } \right) _ { 1 \leqslant i , j \leqslant n } \in \mathcal { M } _ { n } ( \mathbb { R } )$. We define $R ( A ) = \left\{ { } ^ { t } X A X \mid X \in \mathbb { R } ^ { n } , \| X \| = 1 \right\}$.
Let $Q$ be a real orthogonal matrix. Prove that $R ( A ) = R \left( { } ^ { t } Q A Q \right)$.

Let $A = \left( a _ { i j } \right) _ { 1 \leqslant i , j \leqslant n } \in \mathcal { M } _ { n } ( \mathbb { R } )$. We define $R ( A ) = \left\{ { } ^ { t } X A X \mid X \in \mathbb { R } ^ { n } , \| X \| = 1 \right\}$.
We consider the following conditions:
(C1) $\quad \operatorname { Tr } ( A ) \in R ( A )$
(C2) There exists a real orthogonal matrix $Q$ such that the diagonal of the matrix ${ } ^ { t } Q A Q$ is of the form $( \operatorname { Tr } ( A ) , 0 , \ldots , 0 )$
I.F.1) Prove that condition (C2) implies condition (C1).
I.F.2) We assume that $x \in R ( A )$.
Prove that there exists an orthogonal matrix $Q _ { 1 }$ such that $${ } ^ { t } Q _ { 1 } A Q _ { 1 } = \left( \begin{array} { c c } x & L \\ C & B \end{array} \right)$$ where $B$ is a matrix of format $( n - 1 , n - 1 )$ $\left( B \in \mathcal { M } _ { n - 1 } ( \mathbb { R } ) \right)$, $C$ a column vector with $n - 1$ elements $\left( C \in \mathcal { M } _ { n - 1,1 } ( \mathbb { R } ) \right)$ and $L$ a row vector with $n - 1$ elements $\left( L \in \mathcal { M } _ { 1 , n - 1 } ( \mathbb { R } ) \right)$.
I.F.3) Prove that if the matrix $A$ is symmetric then so is the matrix $B$ above.
I.F.4) Prove that $\operatorname { Tr } ( A ) = \operatorname { Tr } \left( { } ^ { t } Q _ { 1 } A Q _ { 1 } \right)$.
I.F.5) Deduce that if $A$ is symmetric, condition (C1) implies condition (C2).
One may reason by induction on $n$.

Throughout this part $A$ and $B$ denote real symmetric matrices of $\mathcal { M } _ { 2 } ( \mathbb { R } )$. We denote by $\lambda _ { 1 } \leqslant \lambda _ { 2 }$ (resp. $\mu _ { 1 } \leqslant \mu _ { 2 }$) the eigenvalues of $A$ (resp. $B$). We define $R ( A ) = \left\{ { } ^ { t } X A X \mid X \in \mathbb { R } ^ { n } , \| X \| = 1 \right\}$.
Prove that $R ( A ) = \left[ \lambda _ { 1 } , \lambda _ { 2 } \right]$.

Throughout this part $A$ and $B$ denote real symmetric matrices of $\mathcal { M } _ { 2 } ( \mathbb { R } )$. We denote by $\lambda _ { 1 } \leqslant \lambda _ { 2 }$ (resp. $\mu _ { 1 } \leqslant \mu _ { 2 }$) the eigenvalues of $A$ (resp. $B$).
Prove that $\operatorname { Tr } ( A B ) \leqslant \lambda _ { 1 } \mu _ { 1 } + \lambda _ { 2 } \mu _ { 2 }$.
One may use an orthogonal matrix $P$ such that ${ } ^ { t } P B P$ is a diagonal matrix, to obtain ${ } ^ { t } P A P = A ^ { \prime } = \left( a _ { i j } ^ { \prime } \right)$ with $\operatorname { Tr } ( A ) = \lambda _ { 1 } + \lambda _ { 2 } = a _ { 11 } ^ { \prime } + a _ { 22 } ^ { \prime }$.

Throughout this part $A$ and $B$ denote real symmetric matrices of $\mathcal { M } _ { 2 } ( \mathbb { R } )$. We set $$A = \left( \begin{array} { l l } a & b \\ b & d \end{array} \right)$$ and assume $A \geqslant 0$ (i.e., all eigenvalues of $A$ are $\geqslant 0$).
II.D.1) Prove that $\operatorname { det } ( A ) \geqslant 0$.
II.D.2) Prove that ${ } ^ { t } X A X \geqslant 0$ for every vector $X$.
II.D.3) Prove that $a \geqslant 0$ and $d \geqslant 0$.
II.D.4) Let $S \in \mathcal { M } _ { 2 } ( \mathbb { R } )$ be symmetric. Prove that: $$S \geqslant 0 \quad \text { if and only if } \quad ( \operatorname { Tr } ( S ) \geqslant 0 \text { and } \operatorname { det } ( S ) \geqslant 0 )$$

Throughout this part $A$ and $B$ denote real symmetric matrices of $\mathcal { M } _ { 2 } ( \mathbb { R } )$. We set $$A = \left( \begin{array} { l l } a _ { 1 } & b _ { 1 } \\ b _ { 1 } & d _ { 1 } \end{array} \right) \quad B = \left( \begin{array} { l l } a _ { 2 } & b _ { 2 } \\ b _ { 2 } & d _ { 2 } \end{array} \right)$$ We assume in this section that $A \geqslant 0$ and $B \geqslant 0$.
II.E.1) By applying the Cauchy-Schwarz inequality to the vectors $( b _ { 1 } , \sqrt { \operatorname { det } A }$ ) and $( b _ { 2 } , \sqrt { \operatorname { det } B }$ ), prove that $$b _ { 1 } b _ { 2 } \leqslant \sqrt { a _ { 1 } a _ { 2 } d _ { 1 } d _ { 2 } } - \sqrt { \operatorname { det } A \operatorname { det } B }$$
II.E.2) By computing $\operatorname { det } ( A + B ) - \operatorname { det } A - \operatorname { det } B$, deduce that $$\operatorname { det } ( A + B ) \geqslant \operatorname { det } ( A ) + \operatorname { det } ( B ) + 2 \sqrt { \operatorname { det } ( A ) \operatorname { det } ( B ) }$$

Throughout this part $A$ and $B$ denote real symmetric matrices of $\mathcal { M } _ { 2 } ( \mathbb { R } )$. We set $$A = \left( \begin{array} { l l } a _ { 1 } & b _ { 1 } \\ b _ { 1 } & d _ { 1 } \end{array} \right) \quad B = \left( \begin{array} { l l } a _ { 2 } & b _ { 2 } \\ b _ { 2 } & d _ { 2 } \end{array} \right)$$ We assume $A \geqslant 0$ and $B \geqslant 0$, $\operatorname { det } A \operatorname { det } B \neq 0$ and $b _ { 1 } b _ { 2 } \neq 0$.
II.F.1) Prove that we have equality in the formula of question II.E.2 if and only if the vectors $( a _ { 1 } , d _ { 1 } )$ and $( a _ { 2 } , d _ { 2 } )$ are linearly dependent, as well as the vectors $( b _ { 1 } , \sqrt { \operatorname { det } A }$ ) and $( b _ { 2 } , \sqrt { \operatorname { det } B }$ ).
II.F.2) Prove then that we have equality in the formula of question II.E.2 if and only if the matrices $A$ and $B$ are proportional ($A = \lambda B$ for some $\lambda \in \mathbb { R }$, $\lambda > 0$).

We consider the following relation on the set of real symmetric matrices of format $(2,2)$: we say that $S \leqslant S ^ { \prime }$ if and only if the symmetric matrix $S ^ { \prime } - S$ satisfies $S ^ { \prime } - S \geqslant 0$ (i.e., all eigenvalues of $S' - S$ are $\geqslant 0$).
Prove that the relation $\leqslant$ above is indeed an order relation on real symmetric matrices of format $(2,2)$.

We consider a sequence $\left( A _ { n } \right) _ { n \geqslant 0 }$ $$A _ { n } = \left( \begin{array} { l l } a _ { n } & b _ { n } \\ b _ { n } & d _ { n } \end{array} \right)$$ of symmetric matrices of $\mathcal { M } _ { 2 } ( \mathbb { R } )$. We say that $S \leqslant S'$ if and only if $S' - S \geqslant 0$. We assume that the sequence $\left( A _ { n } \right) _ { n \geqslant 0 }$ is increasing and bounded for this order relation.
II.H.1) Prove that for every vector $X$, the sequence $\left( { } ^ { t } X A _ { n } X \right) _ { n \geqslant 0 }$ is increasing and bounded.
II.H.2) Prove that the sequences $\left( a _ { n } \right) _ { n \geqslant 0 }$ and $\left( d _ { n } \right) _ { n \geqslant 0 }$ are increasing and bounded.
II.H.3) By considering the vector $X = ( 1,1 )$, prove that the sequence of matrices $\left( A _ { n } \right) _ { n \geqslant 0 }$ is convergent in $\mathcal { M } _ { 2 } ( \mathbb { R } )$, that is, the sequences $\left( a _ { n } \right) _ { n \geqslant 0 }$, $\left( b _ { n } \right) _ { n \geqslant 0 }$ and $\left( d _ { n } \right) _ { n \geqslant 0 }$ are convergent in $\mathbb { R }$.

In this part all matrices are of format $(n, n)$, where $n$ is an integer greater than or equal to 2. We say that a real symmetric matrix is positive definite if and only if all its eigenvalues are strictly positive.
Let $A$ be a positive definite symmetric matrix.
Prove that there exists an invertible matrix $Y$ such that $A = { } ^ { t } Y Y$.

In this part all matrices are of format $(n, n)$, where $n$ is an integer greater than or equal to 2. We say that a real symmetric matrix is positive definite if and only if all its eigenvalues are strictly positive.
Let $A$ be a positive definite symmetric matrix and $B$ a symmetric matrix.
Prove that there exists an invertible matrix $T$ such that: $${ } ^ { t } T A T = I _ { n } \quad \text { and } \quad { } ^ { t } T B T = D$$ where $I _ { n }$ denotes the identity matrix and $D$ a diagonal matrix.

In this part all matrices are of format $(n, n)$, where $n$ is an integer greater than or equal to 2. We say that a real symmetric matrix is positive definite if and only if all its eigenvalues are strictly positive.
Let $A$ and $B$ be two positive definite symmetric matrices.
III.C.1) Prove that: $\operatorname { det } \left( I _ { n } + B \right) \geqslant 1 + \operatorname { det } B$.
III.C.2) Deduce that: $\operatorname { det } ( A + B ) \geqslant \operatorname { det } A + \operatorname { det } B$.

Let $u$ be an endomorphism of $\mathbb{R}^n$. Show that $u$ is self-adjoint positive definite if and only if its matrix in any orthonormal basis belongs to $\mathcal{S}_n^{++}(\mathbb{R})$.

Show that if $S \in \mathcal{S}_n^{++}(\mathbb{R})$, then $S$ is invertible and $S^{-1} \in \mathcal{S}_n^{++}(\mathbb{R})$.

In this question, $u$ denotes an endomorphism of $\mathbb{R}^n$ that is self-adjoint positive definite. We propose to prove that there exists a unique endomorphism $v$ of $\mathbb{R}^n$ that is self-adjoint, positive definite, such that $v^2 = u$.
Let $v$ be an endomorphism of $\mathbb{R}^n$, self-adjoint positive definite and satisfying $v^2 = u$, and let $\lambda$ be an eigenvalue of $u$. Show that $v$ induces an endomorphism of $\operatorname{Ker}(u - \lambda \mathrm{Id})$ which we shall determine.

In this question, $u$ denotes an endomorphism of $\mathbb{R}^n$ that is self-adjoint positive definite. We propose to prove that there exists a unique endomorphism $v$ of $\mathbb{R}^n$ that is self-adjoint, positive definite, such that $v^2 = u$.
Deduce $v$, then conclude.

In this question, $u$ denotes an endomorphism of $\mathbb{R}^n$ that is self-adjoint positive definite. We propose to prove that there exists a unique endomorphism $v$ of $\mathbb{R}^n$ that is self-adjoint, positive definite, such that $v^2 = u$.
Show that there exists a polynomial $Q$ with real coefficients such that $v = Q(u)$.

Let $A \in \mathrm{GL}_n(\mathbb{R})$. Show that ${}^t A A \in \mathcal{S}_n^{++}(\mathbb{R})$.

Let $A \in \mathrm{GL}_n(\mathbb{R})$. Deduce that there exists a unique pair $(O, S) \in \mathrm{O}(n) \times \mathcal{S}_n^{++}(\mathbb{R})$ such that $A = OS$.

Let $A \in \mathrm{GL}_n(\mathbb{R})$. Determine the matrices $O$ and $S$ when $A = \left(\begin{array}{ccc} 3 & 0 & -1 \\ \sqrt{2}/2 & 3\sqrt{2} & -3\sqrt{2}/2 \\ -\sqrt{2}/2 & 3\sqrt{2} & 3\sqrt{2}/2 \end{array}\right)$.

Show that $\mathrm{O}(n)$ is a compact subset of $\mathcal{M}_n(\mathbb{R})$.

Show that $\mathcal{S}_n^+(\mathbb{R})$ is a closed subset of $\mathcal{M}_n(\mathbb{R})$.

Show that $\mathrm{GL}_n(\mathbb{R})$ is a dense subset of $\mathcal{M}_n(\mathbb{R})$.