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.

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 $A = \left(\begin{array}{ccc} 3 & 0 & -1 \\ \sqrt{2}/2 & 3\sqrt{2} & -3\sqrt{2}/2 \\ -\sqrt{2}/2 & 3\sqrt{2} & 3\sqrt{2}/2 \end{array}\right)$.

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

Let $A \in \mathcal{M}_n(\mathbb{R})$. Consider the system $$(*) : \left\{\begin{array}{l} {}^t A A + {}^t X X = I_n \\ {}^t A X - {}^t X A = 0_n \end{array}\right.$$
We assume in this question that the eigenvalues of ${}^t A A$ belong to the interval $[0, 1[$.
a) Justify that we can seek the solutions $X$ of $(*)$ in the form $X = UH$, with $U \in \mathrm{O}(n)$ and $H \in \mathcal{S}_n^{++}(\mathbb{R})$.
b) Determine $H$.
c) Show the existence of a solution $X \in \mathrm{GL}_n(\mathbb{R})$ of $(*)$ belonging to $\mathrm{GL}_n(\mathbb{R})$.

Let $f$ be a linear form on $\mathcal{M}_n(\mathbb{R})$, and let $A$ be the unique matrix in $\mathcal{M}_n(\mathbb{R})$ such that $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$. Let $\mu_1, \ldots, \mu_n$ be the $n$ positive eigenvalues of ${}^t A A$ counted with multiplicities.
Show that $M_n = \sup(\{\operatorname{Tr}(D\Omega), \Omega \in \mathrm{O}(n)\})$, where $D$ is the diagonal matrix whose diagonal elements are $\sqrt{\mu_1}, \ldots, \sqrt{\mu_n}$.

Let $M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$ its ordered spectrum. Show that there exists an orthonormal basis $\left(v_{1}, \ldots, v_{n}\right)$ of $\mathbb{R}^{n}$ such that $$M = \sum_{i=1}^{n} m_{i} v_{i}\, {}^{t}v_{i}$$ Such a decomposition of $M$ will be called in the sequel a spectral resolution of $M$.

In this part, we study the case $n = 2$. For two real numbers $u$ and $v$ such that $u \geqslant v$, we denote: $$S(u, v) = \left\{M \in \mathcal{S}_{2}(\mathbb{R}) \mid s^{\downarrow}(M) = (u, v)\right\}.$$ We fix $a_{1} \geqslant a_{2}$ and $b_{1} \geqslant b_{2}$, four real numbers satisfying the relation $$a_{1} - a_{2} \geqslant b_{1} - b_{2}.$$ We seek to identify the set $$\Sigma = \left\{s^{\downarrow}(A + B) \mid A \in S\left(a_{1}, a_{2}\right),\, B \in S\left(b_{1}, b_{2}\right)\right\}.$$
Show that $\Sigma$ is included in a line segment $L$ of length $\sqrt{2}\left(b_{1} - b_{2}\right)$, and determine its endpoints. One may first study the case where $A$ and $B$ are diagonal.

In this part, we study the case $n = 2$. For two real numbers $u$ and $v$ such that $u \geqslant v$, we denote: $$S(u, v) = \left\{M \in \mathcal{S}_{2}(\mathbb{R}) \mid s^{\downarrow}(M) = (u, v)\right\}.$$ We fix $a_{1} \geqslant a_{2}$ and $b_{1} \geqslant b_{2}$, four real numbers satisfying the relation $a_{1} - a_{2} \geqslant b_{1} - b_{2}$, and $$\Sigma = \left\{s^{\downarrow}(A + B) \mid A \in S\left(a_{1}, a_{2}\right),\, B \in S\left(b_{1}, b_{2}\right)\right\}.$$
Show that $$\Sigma = \left\{s^{\downarrow}(A + B) \,\middle|\, A = \left(\begin{array}{cc} a_{1} & 0 \\ 0 & a_{2} \end{array}\right),\, B \in S\left(b_{1}, b_{2}\right)\right\}.$$

In this part, we study the case $n = 2$. For two real numbers $u$ and $v$ such that $u \geqslant v$, we denote: $$S(u, v) = \left\{M \in \mathcal{S}_{2}(\mathbb{R}) \mid s^{\downarrow}(M) = (u, v)\right\}.$$ We fix $b_{1} \geqslant b_{2}$.
Determine a continuous function defined on $[-\pi, \pi]$ whose image equals $S\left(b_{1}, b_{2}\right)$.

In this part, we study the case $n = 2$. For two real numbers $u$ and $v$ such that $u \geqslant v$, we denote: $$S(u, v) = \left\{M \in \mathcal{S}_{2}(\mathbb{R}) \mid s^{\downarrow}(M) = (u, v)\right\}.$$ We fix $a_{1} \geqslant a_{2}$ and $b_{1} \geqslant b_{2}$, four real numbers satisfying the relation $a_{1} - a_{2} \geqslant b_{1} - b_{2}$, and $$\Sigma = \left\{s^{\downarrow}(A + B) \mid A \in S\left(a_{1}, a_{2}\right),\, B \in S\left(b_{1}, b_{2}\right)\right\}.$$ Let $L$ be the line segment identified in question 9.
Show that $\Sigma = L$.

We consider $A \in \mathcal{M}_{n}(\mathbb{R})$ and assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$.
a) Show that there exists a unique matrix $B$ in $\mathcal{S}_{n}^{++}(\mathbb{R})$ such that $B^{2} = A_{s}$.
b) Show that there exists a matrix $Q$ in $\mathcal{A}_{n}(\mathbb{R})$ such that $\operatorname{det}(A) = \operatorname{det}(A_{s})\operatorname{det}(I_{n} + Q)$.
c) Deduce that $\operatorname{det}(A) \geqslant \operatorname{det}(A_{s})$.

We assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$.
a) Show that there exists a unique matrix $B$ in $\mathcal{S}_{n}^{++}(\mathbb{R})$ such that $B^{2} = A_{s}$.
b) Show that there exists a matrix $Q$ in $\mathcal{A}_{n}(\mathbb{R})$ such that $\operatorname{det}(A) = \operatorname{det}\left(A_{s}\right) \operatorname{det}\left(I_{n} + Q\right)$.
c) Deduce that $\operatorname{det}(A) \geqslant \operatorname{det}\left(A_{s}\right)$.

We fix a natural number $p$ greater than or equal to 2. Prove that every invertible matrix in $\mathcal { M } _ { n } ( \mathbb { C } )$ admits at least one $p$-th root.
One may use the fact that every upper triangular and invertible matrix admits at least one upper triangular $p$-th root (as established in V.B.3).

We equip $\mathbb{R}_{n}[X]$ with the inner product defined by $$\langle P, Q \rangle = \int_{-1}^{1} P(x)Q(x)\,dx$$ For $j \in \mathbb{N}$, we define the polynomial $$P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$$ By convention, $P_{0} = 1$. The subspace of $\mathbb{R}_{n}[X]$ formed by even polynomials is denoted $\Pi_{n}$, and that of odd polynomials is denoted $J_{n}$.
(a) Show that the family $\left(P_{j}\right)_{0 \leqslant j \leqslant n}$ is a basis of $\mathbb{R}_{n}[X]$.
(b) Deduce that the family $\left(P_{2j}\right)_{0 \leqslant j \leqslant \frac{n}{2}}$ is a basis of $\Pi_{n}$, while the family $\left(P_{2j+1}\right)_{0 \leqslant j \leqslant \frac{n-1}{2}}$ is a basis of $J_{n}$.

We consider three strictly positive integers $n , p$ and $k$ such that $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ is non-empty. Let $A$ be a matrix of $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$.
Let $S = A A ^ { \mathrm { T } }$ and $\tilde { S } = A ^ { \mathrm { T } } A$.
(a) Verify that $S$ is a symmetric matrix that admits only non-negative eigenvalues and then show that $\operatorname { Im } ( A ) = \operatorname { Im } ( S )$.
(b) Let $u \in \mathbb { R } ^ { n }$ be an eigenvector of $S$ for an eigenvalue $\lambda > 0$ and let $v = A ^ { \mathrm { T } } u / \sqrt { \lambda } \in \mathbb { R } ^ { p }$. Show that $v$ is an eigenvector of $\tilde { S }$ for the eigenvalue $\lambda$ and $\| v \| _ { 2 } = \| u \| _ { 2 }$.

We consider three strictly positive integers $n , p$ and $k$ such that $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ is non-empty. Let $A$ be a matrix of $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$. Let $S = A A ^ { \mathrm { T } }$ and $\tilde { S } = A ^ { \mathrm { T } } A$.
(a) Show that there exist $U \in \mathscr { M } _ { n , k } ( \mathbb { R } )$ and $\Lambda = \operatorname { diag } \left( \lambda _ { 1 } , \ldots , \lambda _ { k } \right) \in \mathscr { M } _ { k } ( \mathbb { R } )$ such that $S = U \Lambda U ^ { \mathrm { T } }$ with $\lambda _ { 1 } \geqslant \ldots \geqslant \lambda _ { k } > 0$ and $U ^ { \mathrm { T } } U = I _ { k }$.
(b) Show that $\operatorname { Im } ( S ) = \operatorname { Im } ( U )$ and that $U U ^ { \mathrm { T } }$ is the matrix of the orthogonal projection onto $\operatorname { Im } ( U )$ in $\mathbb { R } ^ { n }$.
(c) By setting $V = A ^ { \mathrm { T } } U D \in \mathscr { M } _ { p , k } ( \mathbb { R } )$ where $D = \operatorname { diag } \left( 1 / \sqrt { \lambda _ { 1 } } , \ldots , 1 / \sqrt { \lambda _ { k } } \right) \in \mathscr { M } _ { k } ( \mathbb { R } )$, show that $V ^ { \mathrm { T } } V = I _ { k }$ and $\tilde { S } = V \Lambda V ^ { \mathrm { T } }$.

We consider three strictly positive integers $n , p$ and $k$ such that $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ is non-empty. Let $A$ be a matrix of $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$. Using the decompositions from question 6, deduce that $$A = U \Sigma V ^ { \mathrm { T } } ,$$ with $\Sigma = \operatorname { diag } \left( \sqrt { \lambda _ { 1 } } , \ldots , \sqrt { \lambda } _ { k } \right)$.

Let $A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ be a matrix of rank $k$ where $n , p$ and $k$ are strictly positive integers, $k \leqslant \min ( n , p )$. We consider the decomposition $A = U \Sigma V ^ { T }$ constructed in the first part. Let $l \in \mathbb { N } ^ { * }$ and $\widetilde { V } \in \mathscr { M } _ { p , l } ( \mathbb { R } )$ be such that $l < k$ and $\widetilde { V } ^ { \mathrm { T } } \widetilde { V } = I _ { l }$. We denote by $\left( \tilde { v } _ { 1 } , \ldots , \tilde { v } _ { l } \right) \in \left( \mathbb { R } ^ { p } \right) ^ { l }$ the family of columns of $\widetilde { V }$ and by $\left( v _ { 1 } , \ldots , v _ { k } \right) \in \left( \mathbb { R } ^ { p } \right) ^ { k }$ that of columns of $V$.
(a) Verify that $\left\| A - A \widetilde { V } \widetilde { V } ^ { \mathrm { T } } \right\| _ { F } ^ { 2 } = \| A \| _ { F } ^ { 2 } - \left\| A \widetilde { V } \widetilde { V } ^ { \mathrm { T } } \right\| _ { F } ^ { 2 }$.
(b) Show that $$\left\| A \widetilde { V } \widetilde { V } ^ { \mathrm { T } } \right\| _ { F } ^ { 2 } = \sum _ { h = 1 } ^ { k } \left( \lambda _ { h } \sum _ { m = 1 } ^ { l } \left\langle v _ { h } , \tilde { v } _ { m } \right\rangle _ { 2 } ^ { 2 } \right)$$ where $\langle , \rangle _ { 2 }$ denotes the usual inner product on $\mathbb { R } ^ { p }$.

Let $A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ be a matrix of rank $k$ where $n , p$ and $k$ are strictly positive integers, $k \leqslant \min ( n , p )$. We consider the decomposition $A = U \Sigma V ^ { T }$ constructed in the first part. Let $l \in \mathbb { N } ^ { * }$ and $\widetilde { V } \in \mathscr { M } _ { p , l } ( \mathbb { R } )$ be such that $l < k$ and $\widetilde { V } ^ { \mathrm { T } } \widetilde { V } = I _ { l }$. We denote by $\left( \tilde { v } _ { 1 } , \ldots , \tilde { v } _ { l } \right) \in \left( \mathbb { R } ^ { p } \right) ^ { l }$ the family of columns of $\widetilde { V }$ and by $\left( v _ { 1 } , \ldots , v _ { k } \right) \in \left( \mathbb { R } ^ { p } \right) ^ { k }$ that of columns of $V$.
We assume here that $\lambda _ { l } > \lambda _ { l + 1 }$.
(a) For all $l + 1 \leqslant i \leqslant k$ and all $1 \leqslant j \leqslant l$, we set $a _ { i } = \sum _ { m = 1 } ^ { l } \left\langle v _ { i } , \tilde { v } _ { m } \right\rangle _ { 2 } ^ { 2 }$ and $b _ { j } = 1 - \sum _ { m = 1 } ^ { l } \left\langle v _ { j } , \tilde { v } _ { m } \right\rangle _ { 2 } ^ { 2 }$. Show that the $\left( a _ { i } \right)$ and $\left( b _ { j } \right)$ are non-negative real numbers and that we have $\sum _ { i = l + 1 } ^ { k } a _ { i } \leqslant \sum _ { j = 1 } ^ { l } b _ { j }$.
(b) Show that $\left\| A \widetilde { V } \widetilde { V } ^ { \mathrm { T } } \right\| _ { F } ^ { 2 } \leqslant \sum _ { h = 1 } ^ { l } \lambda _ { h }$ and that we have equality if and only if $\operatorname { Vect } \left( \left\{ v _ { 1 } , \ldots , v _ { l } \right\} \right) = \operatorname { Im } ( \widetilde { V } )$ where $\operatorname { Vect } ( X )$ denotes the vector subspace spanned by $X \subset \mathbb { R } ^ { p }$.
(c) Let $M \in \mathscr { M } _ { n , p } ^ { l } ( \mathbb { R } )$. Show that $\| M - A \| _ { F } ^ { 2 } \geqslant \sum _ { h = l + 1 } ^ { k } \lambda _ { h }$ with equality if and only if $M = U _ { * } \Sigma _ { * } V _ { * } ^ { \mathrm { T } }$ where $\Sigma _ { * } = \operatorname { diag } \left( \sqrt { \lambda _ { 1 } } , \ldots , \sqrt { \lambda _ { l } } \right) , U _ { * }$ (resp. $V _ { * }$ ) is the matrix formed by the first $l$ columns of $U$ (resp. of $V$ ).

Let $n , p$ and $k$ be three strictly positive integers such that $k \leqslant \min ( n , p )$. Let $A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ be a matrix of rank $k$ and $( U , \Sigma , V ) \in \mathscr { E }$ such that $A = U \Sigma V ^ { \mathrm { T } }$, $U ^ { \mathrm { T } } U = V ^ { \mathrm { T } } V = I _ { k }$ and $\Sigma$ diagonal with strictly positive diagonal coefficients.
We denote by $T _ { A } = \left\{ \bar { U } \Sigma V ^ { \mathrm { T } } + U \bar { \Sigma } V ^ { \mathrm { T } } + U \Sigma \bar { V } ^ { \mathrm { T } } \mid ( \bar { U } , \bar { \Sigma } , \bar { V } ) \in \mathscr { E } , \bar { U } ^ { \mathrm { T } } U = \bar { V } ^ { \mathrm { T } } V = O _ { k } \right\}$.
(a) Verify that all elements of $T _ { A }$ are tangent vectors to $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ at $A$ and that $T _ { A }$ is a vector subspace of $\mathscr { M } _ { n , p } ( \mathbb { R } )$ whose dimension you will give.
(b) Let $N _ { A } = \left\{ \bar { N } \in \mathscr { M } _ { n , p } ( \mathbb { R } ) \mid \bar { N } ^ { \mathrm { T } } U = O _ { p , k } , \bar { N } V = O _ { n , k } \right\}$. Show that $N _ { A }$ is the orthogonal subspace to $T _ { A }$ in $\mathscr { M } _ { n , p } ( \mathbb { R } )$ for the inner product $\langle , \rangle _ { F }$.

Let $n , p$ and $k$ be three strictly positive integers such that $k \leqslant \min ( n , p )$. Let $A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ be a matrix of rank $k$ and $( U , \Sigma , V ) \in \mathscr { E }$ such that $A = U \Sigma V ^ { \mathrm { T } }$, $U ^ { \mathrm { T } } U = V ^ { \mathrm { T } } V = I _ { k }$ and $\Sigma$ diagonal with strictly positive diagonal coefficients.
Let $\tilde { A } \in \mathscr { M } _ { n , p } ( \mathbb { R } )$. We say that $\tilde { A }$ satisfies condition (C) if $$\text { (C) } \operatorname { Im } \left( \tilde { \mathrm { A } } V V ^ { \mathrm { T } } \right) = \operatorname { Im } ( \tilde { \mathrm { A } } ) \text { and } \operatorname { Im } \left( \tilde { \mathrm { A } } ^ { \mathrm { T } } \mathrm { UU } ^ { \mathrm { T } } \right) = \operatorname { Im } \left( \tilde { \mathrm { A } } ^ { \mathrm { T } } \right)$$
(a) Show that if $\tilde { A }$ satisfies condition (C) then $\operatorname { rg } ( \tilde { A } ) \leqslant k$ and $$\operatorname { Im } \left( \tilde { A } ^ { \mathrm { T } } U U ^ { \mathrm { T } } \right) ^ { \perp } = \operatorname { ker } ( \tilde { A } )$$
(b) Show that there exists $\epsilon > 0$ such that for all $\tilde { A } \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$, the matrix $\tilde { A }$ satisfies condition (C) as soon as $\| \tilde { A } - A \| _ { F } \leqslant \epsilon$.

Let $n , p$ and $k$ be three strictly positive integers such that $k \leqslant \min ( n , p )$. Let $A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ be a matrix of rank $k$ and $( U , \Sigma , V ) \in \mathscr { E }$ such that $A = U \Sigma V ^ { \mathrm { T } }$, $U ^ { \mathrm { T } } U = V ^ { \mathrm { T } } V = I _ { k }$ and $\Sigma$ diagonal with strictly positive diagonal coefficients. Let $T _ { A } = \left\{ \bar { U } \Sigma V ^ { \mathrm { T } } + U \bar { \Sigma } V ^ { \mathrm { T } } + U \Sigma \bar { V } ^ { \mathrm { T } } \mid ( \bar { U } , \bar { \Sigma } , \bar { V } ) \in \mathscr { E } , \bar { U } ^ { \mathrm { T } } U = \bar { V } ^ { \mathrm { T } } V = O _ { k } \right\}$.
Show that $T _ { A }$ is exactly the set of tangent vectors to $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ at $A$.

Let $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$. We denote by $0 < \lambda _ { 1 } < \cdots < \lambda _ { d }$ the $d$ eigenvalues of $A$ (distinct pairwise) and $F _ { 1 } , \ldots , F _ { d }$ the associated eigenspaces. We consider the linear map from $\mathbb { R } ^ { N }$ to $\mathbb { R } ^ { N }$: $$x \mapsto \sum _ { i = 1 } ^ { d } \lambda _ { i } ^ { 1 / 2 } p _ { F _ { i } } ( x )$$ where $p _ { F _ { i } }$ is the orthogonal projection (for the canonical inner product) onto $F _ { i }$. We denote by $A ^ { 1 / 2 }$ the matrix associated with this linear map in the canonical basis.
a) We write $A = U D U ^ { T }$, where $D \in \mathcal { M } _ { N } ( \mathbb { R } )$ is the diagonal matrix containing the eigenvalues of $A$ in increasing order, with their multiplicities, and $U$ an orthogonal matrix. We denote by $D ^ { 1 / 2 }$ the diagonal matrix whose diagonal coefficients are the square roots of those of $D$. Show that $A ^ { 1 / 2 } = U D ^ { 1 / 2 } U ^ { T }$.
b) Show that $A ^ { 1 / 2 } \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$, that $A ^ { 1 / 2 } A ^ { 1 / 2 } = A$, and that $A ^ { 1 / 2 }$ commutes with $A$.
c) Show that, for all $x \in \mathbb { R } ^ { N } , \| x \| _ { A } = \left\| A ^ { 1 / 2 } x \right\|$, where $\| x \| _ { A }$ is the norm defined in question 4.