The matrix $\left( \begin{array} { c c } \pi & \pi \\ 0 & \frac { 22 } { 7 } \end{array} \right)$ is diagonalizable over $\mathbb { C }$.

Let $M _ { n } ( \mathbb { C } )$ denote the set of $n \times n$ matrices over $\mathbb { C }$. Think of $M _ { n } ( \mathbb { C } )$ as the $2 n ^ { 2 }$-dimensional Euclidean space $\mathbb { R } ^ { 2 n ^ { 2 } }$. Show that the set of all diagonalizable $n \times n$ matrices is dense in $M _ { n } ( \mathbb { C } )$.

Let $A$ be a non-zero matrix of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$. Show that the following properties are equivalent:
i. The matrix $A$ is nilpotent;
ii. The spectrum of $A$ is equal to $\{ 0 \}$;
iii. The matrix $A$ is similar to the matrix $\left( \begin{array} { l l } 0 & 1 \\ 0 & 0 \end{array} \right)$.

We assume in this question that $\mathbb { K }$ is equal to $\mathbb { C }$.
Show that two non-zero matrices of $\mathcal { M } _ { 0 } ( 2 , \mathbb { C } )$ are similar if and only if they have the same characteristic polynomial.

We assume in this question that $\mathbb { K }$ is equal to $\mathbb { C }$.
Does the result that two non-zero matrices of $\mathcal { M } _ { 0 } ( 2 , \mathbb { C } )$ are similar if and only if they have the same characteristic polynomial remain true for two non-zero matrices of $\mathcal { M } _ { 0 } ( n , \mathbb { C } )$, with $n \geq 3$?

We assume in this question that $\mathbb { K }$ is equal to $\mathbb { R }$.
Let $A$ be a matrix of $\mathcal { M } _ { 0 } ( 2 , \mathbb { R } )$. We assume that its characteristic polynomial equals $X ^ { 2 } + r ^ { 2 }$, where $r$ is a non-zero real number.
a) Justify the existence of a matrix $P \in GL ( 2 , \mathbb { C } )$ satisfying: $ir H _ { 0 } = P ^ { - 1 } A P$. What is the value of the matrix $A ^ { 2 } + r ^ { 2 } I _ { 2 }$?
b) Let $f$ be the endomorphism of $\mathbb { R } ^ { 2 }$ canonically associated with the matrix $A$, that is, which maps a column vector $u$ of $\mathbb { R } ^ { 2 }$ to the vector $A u$. Let $w$ be a non-zero vector of $\mathbb { R } ^ { 2 }$. Prove that the family $\left( \frac { 1 } { r } f ( w ) , w \right)$ is a basis of $\mathbb { R } ^ { 2 }$, and give the matrix of $f$ in this basis.

We assume in this question that $\mathbb { K }$ is equal to $\mathbb { R }$.
Show that two non-zero matrices of $\mathcal { M } _ { 0 } ( 2 , \mathbb { R } )$ are similar in $\mathcal { M } ( 2 , \mathbb { R } )$ if and only if they have the same characteristic polynomial.

Let $X, H, Y$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ such that $(X, H, Y)$ forms an admissible triple (i.e., $[H,X]=2X$, $[X,Y]=H$, $[H,Y]=-2Y$).
Show using questions II.F and II.C that there exists a matrix $Q \in GL ( 2 , \mathbb { K } )$ satisfying $X = Q X _ { 0 } Q ^ { - 1 }$.

Let $X, H, Y$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ such that $(X, H, Y)$ forms an admissible triple. Prove the identity $( X , H , Y ) = \left( P X _ { 0 } P ^ { - 1 } , P H _ { 0 } P ^ { - 1 } , P Y _ { 0 } P ^ { - 1 } \right)$.

Let $V$ be a $\mathbb { K }$-vector space of finite non-zero dimension. Let $f$ be a diagonalizable endomorphism of $V$ and $W$ a non-zero subspace of $V$ stable under $f$. Show that the endomorphism of $W$ induced by $f$ is diagonalizable.

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 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 $A = OB {}^t O$.

For $p \in \mathbb{N}^*$, we set $$A_p = \left(\begin{array}{rrrrr} 2 & -1 & 0 & \cdots & 0 \\ -1 & 2 & -1 & \ddots & \vdots \\ 0 & -1 & 2 & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & -1 \\ 0 & \cdots & 0 & -1 & 2 \end{array}\right) \in \mathcal{M}_p(\mathbb{R})$$
Show that $A_p$ is diagonalizable, and determine a basis of eigenvectors, specifying for each one the associated eigenvalue.

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 $A, B \in M_p(\mathbb{K})$ be two diagonalizable matrices. We assume that $A$ and $B$ commute.
Show that there exists $P \in GL_p(\mathbb{C})$ such that $P^{-1}AP$ and $P^{-1}BP$ are diagonal.
(We shall study the restrictions of $u_B$ to the eigenspaces of $u_A$.)

We are given $S \in \mathbf{S}_n$ and $R \in \mathbf{O}_n$. Verify that ${}^t R S R$ is symmetric and that it is similar to $S$.

Let $\left(A^{(m)}\right)_{m \in \mathbb{N}}$ be a sequence in $\mathbf{M}_n$. We assume that for all $m \in \mathbb{N}$, the matrix $A^{(m+1)}$ is similar to $A^{(m)}$.
Show that $A^{(m)}$ is similar to $A^{(0)}$.

Show that the matrices that are elements of $O ^ { + } ( 1,1 )$ are diagonalizable and find a matrix $P \in O ( 2 )$ such that, for every matrix $L \in O ^ { + } ( 1,1 )$, the matrix ${ } ^ { t } P L P$ is diagonal.

List the elements of $\mathcal{X}_2' = \mathcal{X}_2 \cap \mathrm{GL}_2(\mathbb{R})$. Specify (by justifying) which ones are diagonalizable over $\mathbb{R}$.

Prove that all elements of $\mathcal{P}_n$ are diagonalizable over $\mathbb{C}$.

Let $S \in \mathcal{S}_{n}(\mathbb{R})$.
a) We assume that $\operatorname{sp}_{\mathbb{R}}(S) \subset [-1,1]$ and that for every eigenvalue $\lambda$ of $S$ in $]-1,1[$, the eigenspace of $S$ associated with $\lambda$ has even dimension. Show that there exists $A \in \mathrm{O}_{n}(\mathbb{R})$ such that $A_{s} = S$.
b) Conversely, show that if there exists $A \in \mathrm{O}_{n}(\mathbb{R})$ such that $A_{s} = S$, then $\operatorname{sp}_{\mathbb{R}}(S) \subset [-1,1]$ and for every eigenvalue $\lambda$ of $S$ in $]-1,1[$, the eigenspace of $S$ associated with $\lambda$ has even dimension.

With the same setup as Q27 (matrices $A$, $B$, $I_q$, $r$, $F_n$), justify that the matrices $A$ and $B$ are diagonalizable over $\mathbb{R}$ and that, for all $n \in \mathbb{N}$, $F_{n} = A^{n} F_{0}$.

Let $A = \left(\begin{array}{cc} a & b \\ c & a \end{array}\right)$ be a Toeplitz matrix of size $2 \times 2$, where $(a, b, c)$ are complex numbers. Discuss, depending on the values of $(a, b, c)$, the diagonalizability of $A$.

Let $M = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ be a matrix of $\mathcal{M}_{2}(\mathbb{C})$. Show that $M$ is similar to a matrix of type $\left(\begin{array}{cc} \alpha & 0 \\ 0 & \beta \end{array}\right)$ or of type $\left(\begin{array}{cc} \alpha & \gamma \\ 0 & \alpha \end{array}\right)$, where $\alpha, \beta$ and $\gamma$ are complex numbers with $\alpha \neq \beta$.