We define the sequences $(u_n)$ and $(v_n)$ on the set $\mathbb{N}$ of natural numbers by: $$u_0 = 0 ; v_0 = 1, \text{ and } \left\{\begin{array}{l} u_{n+1} = \dfrac{u_n + v_n}{2} \\ v_{n+1} = \dfrac{u_n + 2v_n}{3} \end{array}\right.$$
The purpose of this exercise is to study the convergence of sequences $(u_n)$ and $(v_n)$.

1. Calculate $u_1$ and $v_1$.
2. We consider the following algorithm:
   Variables: $u$, $v$ and $w$ real numbers; $N$ and $k$ integers Initialization: $u$ takes the value 0; $v$ takes the value 1 Start of algorithm Enter the value of $N$ For $k$ varying from 1 to $N$ $w$ takes the value $u$ $u$ takes the value $\dfrac{w + v}{2}$ $v$ takes the value $\dfrac{w + 2v}{3}$ End of For Display $u$ Display $v$ End of algorithm
   a. We execute this algorithm by entering $N = 2$. Copy and complete the table given below containing the state of variables during the execution of the algorithm.
   

   $k$	$w$	$u$	$v$
   1
   2

   
   b. For a given number $N$, what do the values displayed by the algorithm correspond to with respect to the situation studied in this exercise?
   
3. For all natural numbers $n$ we define the column vector $X_n$ by $X_n = \binom{u_n}{v_n}$ and the matrix $A$ by $$A = \left(\begin{array}{ll} \dfrac{1}{2} & \dfrac{1}{2} \\ \dfrac{1}{3} & \dfrac{2}{3} \end{array}\right).$$ a. Verify that, for all natural numbers $n$, $X_{n+1} = A X_n$. b. Prove by induction that $X_n = A^n X_0$ for all natural numbers $n$.
   
4. We define matrices $P$, $P'$ and $B$ by $$P = \left(\begin{array}{cc} \dfrac{4}{5} & \dfrac{6}{5} \\ -\dfrac{6}{5} & \dfrac{6}{5} \end{array}\right), \quad P' = \left(\begin{array}{cc} \dfrac{1}{2} & -\dfrac{1}{2} \\ \dfrac{1}{2} & \dfrac{1}{3} \end{array}\right)$$

The complex plane is equipped with an orthonormal coordinate system ($\mathrm{O}; \vec{u}, \vec{v}$). We consider the line $\mathscr{D}$ with equation $$7x - 3y - 1 = 0$$ We define the sequence $(A_{n})$ of points in the plane with coordinates $(x_{n}; y_{n})$ satisfying for all natural integer $n$: $$\left\{ \begin{array}{l} x_{0} = 1 \\ y_{0} = 2 \end{array} \quad \text{and} \quad \left\{ \begin{array}{l} x_{n+1} = -\frac{13}{2} x_{n} + 3 y_{n} \\ y_{n+1} = -\frac{35}{2} x_{n} + 8 y_{n} \end{array} \right. \right.$$
We denote by $M$ the matrix $\left( \begin{array}{cc} \frac{-13}{2} & 3 \\ \frac{-35}{2} & 8 \end{array} \right)$.

(Candidates who have followed the specialization course)
We observe the size of an ant colony every day. For any non-zero natural number $n$, we denote by $u _ { n }$ the number of ants, expressed in thousands, in this population at the end of the $n$-th day. At the beginning of the study the colony has 5000 ants and after one day it has 5100 ants. Thus, we have $u _ { 0 } = 5$ and $u _ { 1 } = 5.1$. We assume that the increase in the size of the colony from one day to the next decreases by $10\%$ each day. In other words, for any natural number $n$,
$$u _ { n + 2 } - u _ { n + 1 } = 0.9 \left( u _ { n + 1 } - u _ { n } \right) .$$

1. Prove that under these conditions, $u _ { 2 } = 5.19$.
2. For any natural number $n$, we set $V _ { n } = \binom { u _ { n + 1 } } { u _ { n } }$ and $A = \left( \begin{array} { c c } 1.9 & - 0.9 \\ 1 & 0 \end{array} \right)$. a. Prove that, for any natural number $n$, we have $V _ { n + 1 } = A V _ { n }$.

We then admit that, for any natural number $n$, $V _ { n } = A ^ { n } V _ { 0 }$. b. We set $P = \left( \begin{array} { c c } 0.9 & 1 \\ 1 & 1 \end{array} \right)$. We admit that the matrix $P$ is invertible.
Using a calculator, determine the matrix $P ^ { - 1 }$. By detailing the calculations, determine the matrix $D$ defined by $D = P ^ { - 1 } A P$. c. Prove by induction that, for any natural number $n$, we have $A ^ { n } = P D ^ { n } P ^ { - 1 }$. For any natural number $n$, we admit that
$$A ^ { n } = \left( \begin{array} { c c } - 10 \times 0.9 ^ { n + 1 } + 10 & 10 \times 0.9 ^ { n + 1 } - 9 \\ - 10 \times 0.9 ^ { n } + 10 & 10 \times 0.9 ^ { n } - 9 \end{array} \right) .$$
d. Deduce that, for any natural number $n$, $u _ { n } = 6 - 0.9 ^ { n }$.

1. Calculate the size of the colony at the end of the $10 ^ { \mathrm { th } }$ day. Round the result to the nearest ant.
2. Calculate the limit of the sequence $(u _ { n })$. Interpret this result in context.

A $5 \times 5$ real matrix has an eigenvector in $\mathbb{R}^5$.

There is $2 \times 2$ real matrix with characteristic polynomial $x ^ { 2 } + 1$.

Let $A : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ be a linear transformation with eigenvalues $\frac { 2 } { 3 }$ and $\frac { 9 } { 5 }$. Then, there exists a non-zero vector $v \in \mathbb { R } ^ { 2 }$ such that
(a) $\| A v \| > 2 \| v \|$;
(b) $\| A v \| < \frac { 1 } { 2 } \| v \|$;
(c) $\| A v \| = \| v \|$;
(d) $A v = 0$;

Let $v$ be a (fixed) unit vector in $\mathbb { R } ^ { 3 }$. (We think of elements of $\mathbb { R } ^ { n }$ as column vectors.) Let $M = I _ { 3 } - 2 v v ^ { t }$. Pick the correct statement(s) from below.
(A) $O$ is an eigenvalue of $M$.
(B) $M ^ { 2 } = I _ { 3 }$.
(C) 1 is an eigenvalue of $M$.
(D) The eigenspace for the eigenvalue $-1$ is 2-dimensional.

Let $A \in \mathrm { GL } ( 3 , \mathbb { Q } )$ with $A ^ { t } A = I _ { 3 }$. Assume that $$A \left[ \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right] = \lambda \left[ \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right]$$ for some $\lambda \in \mathbb { C }$.
(A) Determine the possible values of $\lambda$.
(B) Determine $x + y + z$ where $x , y , z$ is given by $$\left[ \begin{array} { l } x \\ y \\ z \end{array} \right] = A \left[ \begin{array} { c } 1 \\ - 1 \\ 0 \end{array} \right]$$

Consider the real matrix
$$A = \left( \begin{array} { l l } \lambda & 2 \\ 3 & 5 \end{array} \right)$$
Assume that $-1$ is an eigenvalue of $A$. Which of the following are true?
(A) The other eigenvalue is in $\mathbb { C } \backslash \mathbb { R }$.
(B) $A + I _ { 2 }$ is singular.
(C) $\lambda = 1$.
(D) Trace of $A$ is 5.

For the system of linear equations in $x , y$ $$\left( \begin{array} { c c } 1 & a - 2 \\ 2 & - 1 \end{array} \right) \binom { x } { y } = 3 \binom { x } { y }$$ Find the value of the constant $a$ such that the system has a solution other than $x = 0 , y = 0$. [3 points]

We consider the matrix with real coefficients $C \in \mathscr { M } _ { 7 } ( \mathbb { R } )$
$$C = \left( \begin{array} { l l l l l l l } 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \end{array} \right)$$
Let $\Phi$ be the matrix in the basis $(f_1, f_2, f_3)$ of the endomorphism $\varphi$ of $F$ induced by $c$ (as determined in I.B).
In this question, we propose to calculate the spectrum of $\Phi$ without calculating its characteristic polynomial. I.C.1) Why is 1 an eigenvalue of $\Phi$? I.C.2) Can we deduce from the sole calculation of the trace of $\Phi$ that $\Phi$ is diagonalizable in $\mathscr{M}_3(\mathbb{C})$? I.C.3) Calculate $\Phi^2$. Using the additional information obtained by calculating the trace of $\Phi^2$, determine the spectrum of $\Phi$. Is the matrix $\Phi$ diagonalizable in $\mathscr{M}_3(\mathbb{R})$?

We consider the matrix with real coefficients $C \in \mathscr { M } _ { 7 } ( \mathbb { R } )$
$$C = \left( \begin{array} { l l l l l l l } 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \end{array} \right)$$
I.D.1) Deduce from the previous questions the spectrum of $C$. Specify the multiplicity order of the eigenvalues. I.D.2) Is the matrix $C$ diagonalizable over $\mathbb{C}$? over $\mathbb{R}$? If yes, indicate a diagonal matrix similar to $C$.

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
We assume that $h^{-1} \circ h'$ has $n$ distinct eigenvalues. Show that there exists a basis of $E$ orthogonal for both $q$ and $q'$.

Let $V$ be a $\mathbb { K }$-vector space of finite non-zero dimension. Let $f$ and $g$ be two endomorphisms of $V$ that commute, that is, such that $f \circ g = g \circ f$. Show that the eigenspaces of $f$ are stable under $g$.

Let $V$ be a $\mathbb { K }$-vector space of finite non-zero dimension. Let $I$ be a non-empty set and let $\left\{ f _ { i } \mid i \in I \right\}$ be a family of diagonalizable endomorphisms of $V$ commuting pairwise. Show that there exists a basis of $V$ in which the matrices of the endomorphisms $f _ { i }$, for $i \in I$, are diagonal. Hint: one may first treat the case where all the endomorphisms $f _ { i }$ are homotheties, then reason by induction on the dimension of $V$.

Let $\mathcal { A }$ be a non-zero vector subspace of $\mathcal { M } ( n , \mathbb { K } )$ stable by bracket, and let $\mathcal { E }$ be the intersection of $\mathcal { A }$ and $\mathcal { D } ( n , \mathbb { K } )$. Let $H$ be an element of $\mathcal { E }$.
a) Calculate the image under $\Phi _ { H }$ of the canonical basis of $\mathcal { M } ( n , \mathbb { K } )$. Deduce that $\Phi _ { H }$ is a diagonalisable endomorphism of $\mathcal { M } ( n , \mathbb { K } )$.
b) Show that there exists a basis of $\mathcal { A }$ in which the matrices of the endomorphisms of $\mathcal { A }$ induced by the $\Phi _ { H }$, for $H \in \mathcal { E }$, are diagonal.

Let $\mathcal { A }$ be a non-zero vector subspace of $\mathcal { M } ( n , \mathbb { K } )$ stable by bracket, and let $\mathcal { E }$ be the intersection of $\mathcal { A }$ and $\mathcal { D } ( n , \mathbb { K } )$. For every map $\lambda$ from $\mathcal { E }$ to $\mathbb { K }$, we set: $$\mathcal { A } _ { \lambda } = \left\{ M \in \mathcal { A } \mid \Phi _ { H } ( M ) = \lambda ( H ) M \text { for all } H \in \mathcal { E } \right\}$$
Let $\lambda$ be a map from $\mathcal { E }$ to $\mathbb { K }$.
a) Show that $\mathcal { A } _ { \lambda }$ is a vector subspace of $\mathcal { A }$.
b) Show that if $\mathcal { A } _ { \lambda }$ is not reduced to $\{ 0 \}$, then $\lambda$ is a linear form on $\mathcal { E }$.

Let $n \in \mathbb{N}^*$ and $A \in \mathcal{S}_n(\mathbb{R})$. Show that $A$ is positive if and only if all its eigenvalues are positive.

Let $n \in \mathbb{N}^*$ and $A \in \mathcal{S}_n(\mathbb{R})$. Show that $A$ is positive definite if and only if all its eigenvalues are strictly positive.

For $n \in \mathbb{N}^*$, $A \in \mathcal{S}_n(\mathbb{R})$ and $i \in \llbracket 1; n \rrbracket$, we denote by $A^{(i)}$ the square matrix of order $i$ extracted from $A$, consisting of the first $i$ rows and the first $i$ columns of $A$.
Let $A \in \mathcal{S}_n(\mathbb{R})$. We assume that $A$ is positive definite.
For all $i \in \llbracket 1; n \rrbracket$, show that the matrix $A^{(i)}$ is positive definite and deduce that $\operatorname{det}\left(A^{(i)}\right) > 0$.

For $n \in \mathbb{N}^*$, $A \in \mathcal{S}_n(\mathbb{R})$ and $i \in \llbracket 1; n \rrbracket$, we denote by $A^{(i)}$ the square matrix of order $i$ extracted from $A$, consisting of the first $i$ rows and the first $i$ columns of $A$. For all $n \in \mathbb{N}^*$, we say that a matrix $A$ of $\mathcal{S}_n(\mathbb{R})$ satisfies property $\mathcal{P}_n$ if $\operatorname{det}\left(A^{(i)}\right) > 0$ for all $i \in \llbracket 1; n \rrbracket$.
Let $n \in \mathbb{N}^*$. We assume that any matrix of $\mathcal{S}_n(\mathbb{R})$ satisfying property $\mathcal{P}_n$ is positive definite. We consider a matrix $A$ of $\mathcal{S}_{n+1}(\mathbb{R})$ satisfying property $\mathcal{P}_{n+1}$ and we assume by contradiction that $A$ is not positive definite.
a) Show then that $A$ admits two linearly independent eigenvectors associated with eigenvalues (not necessarily distinct) that are strictly negative.
b) Deduce that there exists $X \in \mathcal{M}_{n+1,1}(\mathbb{R})$ whose last component is zero and such that ${}^t X A X < 0$.
c) Conclude.

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$
Prove that $H_n$ is invertible, then that $\operatorname{det}\left(H_n^{-1}\right)$ is an integer.

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$
Demonstrate that $H_n$ admits $n$ real eigenvalues (counted with their multiplicity) that are strictly positive.

We set $\xi_i^j = \begin{cases}0 & \text{if } i = j \\ 1 & \text{otherwise}\end{cases}$ and $N_k = (m_{ij} + k\xi_i^j)$ with $k$ a strictly positive real number.
Let $A \in \mathcal{M}_n(\mathbb{R})$. Show that the hyperplane $\mathcal{H}$ with normal vector $Z$ (and equation $x_1 + \cdots + x_n = 0$) is stable under the canonical endomorphism associated with the matrix $\Psi(A)$.

We set $\xi_i^j = \begin{cases}0 & \text{if } i = j \\ 1 & \text{otherwise}\end{cases}$ and $N_k = (m_{ij} + k\xi_i^j)$ with $k$ a strictly positive real number.
Show that there exists a minimal real $k_0$ that we will specify as a function of the eigenvalues of $\Psi(M)$, such that the matrix $\Psi(N_{k_0})$ has non-negative eigenvalues.