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

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 $\varphi$ be the map from $\mathrm{O}(n) \times \mathcal{S}_n^{++}(\mathbb{R})$ to $\mathrm{GL}_n(\mathbb{R})$ defined by $\varphi(O, S) = OS$ for every pair $(O, S)$ of $\mathrm{O}(n) \times \mathcal{S}_n^{++}(\mathbb{R})$.
Show that $\varphi$ is bijective, continuous, and that its inverse is continuous.

In this part, $A$ and $B$ denote two matrices of $\mathcal{M}_n(\mathbb{R})$. We assume that there exists a square matrix $U$ of size $n$, invertible, with complex coefficients, such that $U {}^t\bar{U} = I_n$ and $A = UBU^{-1}$, where $\bar{U}$ denotes the matrix whose coefficients are the conjugates of those of $U$.
Justify that ${}^t A = U({}^t B)U^{-1}$.

In this part, $A$ and $B$ denote two matrices of $\mathcal{M}_n(\mathbb{R})$. We assume that there exists a square matrix $U$ of size $n$, invertible, with complex coefficients, such that $U {}^t\bar{U} = I_n$ and $A = UBU^{-1}$, where $\bar{U}$ denotes the matrix whose coefficients are the conjugates of those of $U$.
We propose to show that there exists a matrix $P \in \mathrm{GL}_n(\mathbb{R})$ such that $A = PBP^{-1}$ and ${}^t A = P {}^t B P^{-1}$. For this, we denote by $X$ and $Y$ the matrices of $\mathcal{M}_n(\mathbb{R})$ such that $U = X + \mathrm{i}Y$.
a) Show that there exists $\mu \in \mathbb{R}$ such that $X + \mu Y \in \mathrm{GL}_n(\mathbb{R})$.
b) Show that $AX = XB$ and $AY = YB$.
c) Conclude.

In this part, $A$ and $B$ denote two matrices of $\mathcal{M}_n(\mathbb{R})$. We assume that there exists a matrix $P \in \mathrm{GL}_n(\mathbb{R})$ such that $A = PBP^{-1}$ and ${}^t A = P {}^t B P^{-1}$. We write $P$ in the form $P = OS$, with $O \in \mathrm{O}(n)$ and $S \in \mathcal{S}_n^{++}(\mathbb{R})$.
a) Show that $BS^2 = S^2 B$, then that $BS = SB$.
b) Deduce that there exists $O \in \mathrm{O}(n)$ such that $A = OB {}^t O$.

Let $A \in \mathcal{M}_n(\mathbb{R})$. We propose to give a necessary and sufficient condition for the existence of a solution $X \in \mathrm{GL}_n(\mathbb{R})$ to the system $$(*) : \left\{\begin{array}{l} {}^t A A + {}^t X X = I_n \\ {}^t A X - {}^t X A = 0_n \end{array}\right.$$
Show that if the system $(*)$ admits a solution in $\mathrm{GL}_n(\mathbb{R})$, then the eigenvalues of ${}^t A A$ belong to the interval $[0, 1[$.

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 $\mathcal{M}_n(\mathbb{R})$.
Show that there exists a unique matrix $A \in \mathcal{M}_n(\mathbb{R})$ such that $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$.

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 $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$.
Justify the existence of $M_n = \sup(\{f(O), O \in \mathrm{O}(n)\})$.

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 $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$.
Justify that ${}^t A A$ admits $n$ positive eigenvalues $\mu_1, \ldots, \mu_n$, counted with multiplicities.

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 $f$ 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, and $D$ the diagonal matrix whose diagonal elements are $\sqrt{\mu_1}, \ldots, \sqrt{\mu_n}$.
Deduce that $M_n = \sum_{k=1}^n \sqrt{\mu_k}$.

In this question, $f$ denotes the linear form defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \sum_{j=1}^n \sum_{i=j}^n m_{i,j}$.
Determine the matrix $A$ such that $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$.

In this question, $f$ denotes the linear form defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \sum_{j=1}^n \sum_{i=j}^n m_{i,j}$, and $A$ is the matrix such that $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$.
Show that $$A^{-1} = \left(\begin{array}{rrrrr} 1 & -1 & 0 & \cdots & 0 \\ 0 & 1 & -1 & \ddots & \vdots \\ \vdots & \ddots & 1 & \ddots & 0 \\ \vdots & & \ddots & \ddots & -1 \\ 0 & \cdots & \cdots & 0 & 1 \end{array}\right)$$

In this question, $f$ denotes the linear form defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \sum_{j=1}^n \sum_{i=j}^n m_{i,j}$, and $A$ is the matrix such that $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$, with $A^{-1}$ as given in IV.C.2.
Determine the eigenvalues of $A^{-1} {}^t A^{-1}$.

Let $n \in \mathbb{N}^*$. Let $A \in M_2(\mathbb{R})$ be an antisymmetric matrix. We set $A = \begin{pmatrix} 0 & -\alpha \\ \alpha & 0 \end{pmatrix}$ where $\alpha \in \mathbb{R}$.
Determine a number $\beta_n \in \mathbb{R}_+^*$ such that $$\frac{1}{\beta_n}\left(I_2 + \frac{1}{n}A\right) \in SO_2(\mathbb{R})$$

Let $n \in \mathbb{N}^*$. Let $A \in M_2(\mathbb{R})$ be an antisymmetric matrix. We set $A = \begin{pmatrix} 0 & -\alpha \\ \alpha & 0 \end{pmatrix}$ where $\alpha \in \mathbb{R}$.
Determine a real number $\theta_n$ such that $$\frac{1}{\beta_n}\left(I_2 + \frac{1}{n}A\right) = \begin{pmatrix} \cos\theta_n & -\sin\theta_n \\ \sin\theta_n & \cos\theta_n \end{pmatrix}$$

Let $n \in \mathbb{N}^*$. Let $A \in M_2(\mathbb{R})$ be an antisymmetric matrix. We set $A = \begin{pmatrix} 0 & -\alpha \\ \alpha & 0 \end{pmatrix}$ where $\alpha \in \mathbb{R}$.
Deduce that $E(A)$ exists and that it is a rotation matrix, whose angle we shall specify.

Let $B \in M_3(\mathbb{R})$ be antisymmetric.
a) Show that $\det B = 0$.
b) Show that $\left(\operatorname{Ker} u_B\right)^\perp$ is stable under $u_B$.
c) Deduce that $B$ has rank 0 or 2.

Let $B \in M_3(\mathbb{R})$ be antisymmetric.
Show that there exist a matrix $P$ of $O_3(\mathbb{R})$ and a real number $\beta$ such that $$B = P \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -\beta \\ 0 & \beta & 0 \end{pmatrix} P^{-1}$$

Let $B \in M_3(\mathbb{R})$ be antisymmetric. Assume that $$B = P \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -\beta \\ 0 & \beta & 0 \end{pmatrix} P^{-1}$$ for some $P \in O_3(\mathbb{R})$ and $\beta \in \mathbb{R}$.
Show that $|\beta| = \frac{\|B\|_2}{\sqrt{2}}$.

Let $B \in M_3(\mathbb{R})$ be antisymmetric.
Show that $E(B)$ exists and is a rotation matrix. Specify the value of its unoriented angle as a function of $\|B\|_2$.

Let $D \in M_p(\mathbb{K})$ be a diagonal matrix.
Show that $E(D)$ exists and that $E(D) \in GL_p(\mathbb{C})$.

Let $A \in M_p(\mathbb{C})$ be nilpotent of order $k$.
Show that $E(A) - I_p$ is nilpotent.