grandes-ecoles

QI.A.1 Sequences and series, recurrence and convergence Monotonicity and boundedness analysis View

Verify that a periodic sequence is bounded.

QI.A.2 Sequences and series, recurrence and convergence True/false or conceptual reasoning about sequences View

What can be said about 1-periodic sequences?

QI.A.3 Sequences and series, recurrence and convergence Proof by induction on sequence properties View

Verify that, if $\left( z _ { k } \right)$ is $p$-periodic, then $\forall n \in \mathbb { N } , \forall k \in \mathbb { N } , z _ { n + k p } = z _ { n }$.

QI.A.4 Sequences and series, recurrence and convergence Convergence proof and limit determination View

What can be said about sequences that are both periodic and convergent?

QI.B.1 Matrices Matrix Norm, Convergence, and Inequality View

Verify the following property: $\forall ( A , B ) \in \left( \mathcal { M } _ { n } ( \mathbb { C } ) \right) ^ { 2 } \quad \| A B \| _ { 0 } \leqslant n \| A \| _ { 0 } \cdot \| B \| _ { 0 }$

QI.B.2 Matrices Matrix Norm, Convergence, and Inequality View

Verify the following property: $\forall A \in \mathcal { M } _ { n } ( \mathbb { C } ) , \forall Y \in \mathbb { C } ^ { n } \quad \| A Y \| _ { \infty } \leqslant n \| A \| _ { 0 } \cdot \| Y \| _ { \infty }$

QII.A.1 Sequences and series, recurrence and convergence Closed-form expression derivation View

In this subsection II.A, $a$ is a nonzero real number. We denote by Sol(II.1) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad z _ { k + 1 } + a z _ { k } + z _ { k - 1 } = 0$$ Give the general form of sequences belonging to $\operatorname { Sol } ($ II.1 $)$ as a function of the complex roots $r _ { 1 }$ and $r _ { 2 }$ of the equation $r ^ { 2 } + a r + 1 = 0$. What are $r _ { 1 } + r _ { 2 }$ and $r _ { 1 } r _ { 2 }$?

QII.A.2 Sequences and series, recurrence and convergence Monotonicity and boundedness analysis View

In this subsection II.A, $a$ is a nonzero real number. We denote by Sol(II.1) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad z _ { k + 1 } + a z _ { k } + z _ { k - 1 } = 0$$ Show that if $| a | > 2$, the zero sequence is the only periodic solution of (II.1).

QII.A.3 Sequences and series, recurrence and convergence Monotonicity and boundedness analysis View

In this subsection II.A, $a$ is a nonzero real number. We denote by Sol(II.1) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad z _ { k + 1 } + a z _ { k } + z _ { k - 1 } = 0$$ Show that if $a = - 2$ then, (II.1) admits infinitely many constant solutions and infinitely many unbounded solutions.

QII.A.4 Sequences and series, recurrence and convergence Monotonicity and boundedness analysis View

In this subsection II.A, $a$ is a nonzero real number. We denote by Sol(II.1) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad z _ { k + 1 } + a z _ { k } + z _ { k - 1 } = 0$$ Show that if $a = + 2$ then, (II.1) admits infinitely many 2-periodic solutions and infinitely many unbounded solutions.

QII.A.5 Sequences and series, recurrence and convergence Convergence proof and limit determination View

In this subsection II.A, $a$ is a nonzero real number. We denote by Sol(II.1) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad z _ { k + 1 } + a z _ { k } + z _ { k - 1 } = 0$$ Suppose in this question that $p$ is an integer greater than or equal to 3. Give a value of $a \in ] - 2,2 [$ for which all solutions of equation (II.1) are $p$-periodic.

QII.B.1 Matrices Linear Transformation and Endomorphism Properties View

We assume that $p$ is an integer greater than or equal to 2, that $\left( a _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( b _ { k } \right) _ { k \in \mathbb { N } }$ are two sequences of real numbers that are $p$-periodic and that $\forall k \in \mathbb { N } , b _ { k } \neq 0$. We denote by Sol(II.2) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ that satisfy the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$ Justify that the application $\left. \Psi : \left\lvert \, \begin{array} { l l l } \operatorname { Sol } ( \mathrm { II } .2 ) & \rightarrow & \mathbb { C } ^ { 2 } \\ \left( z _ { k } \right) _ { k \in \mathbb { N } } & \mapsto & \left( z _ { 0 } \right) \\ z _ { 1 } \end{array} \right. \right)$ is an isomorphism of $\mathbb { C }$-vector spaces.

QII.B.2 Sequences and series, recurrence and convergence Proof by induction on sequence properties View

We assume that $p$ is an integer greater than or equal to 2, that $\left( a _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( b _ { k } \right) _ { k \in \mathbb { N } }$ are two sequences of real numbers that are $p$-periodic and that $\forall k \in \mathbb { N } , b _ { k } \neq 0$. We denote by Sol(II.2) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ that satisfy the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$ We fix $\left( y _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( z _ { k } \right) _ { k \in \mathbb { N } }$, two solution sequences of (II.2). We set for all $k \in \mathbb { N } , W _ { k } = b _ { k } \left( y _ { k } z _ { k + 1 } - z _ { k } y _ { k + 1 } \right)$. Show that the sequence $\left( W _ { k } \right) _ { k \in \mathbb { N } }$ is constant.

QII.B.3 Sequences and series, recurrence and convergence Proof by induction on sequence properties View

We assume that $p$ is an integer greater than or equal to 2, that $\left( a _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( b _ { k } \right) _ { k \in \mathbb { N } }$ are two sequences of real numbers that are $p$-periodic and that $\forall k \in \mathbb { N } , b _ { k } \neq 0$. We denote by Sol(II.2) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ that satisfy the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$ We fix $\left( y _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( z _ { k } \right) _ { k \in \mathbb { N } }$, two solution sequences of (II.2), and set for all $k \in \mathbb { N } , W _ { k } = b _ { k } \left( y _ { k } z _ { k + 1 } - z _ { k } y _ { k + 1 } \right)$. Show that the two sequences $\left( y _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ form a basis of $\operatorname { Sol } ($ II.2 $)$ if and only if $W _ { 0 } \neq 0$.

QII.C Matrices Matrix Power Computation and Application View

We assume that $p$ is an integer greater than or equal to 2, that $\left( a _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( b _ { k } \right) _ { k \in \mathbb { N } }$ are two sequences of real numbers that are $p$-periodic and that $\forall k \in \mathbb { N } , b _ { k } \neq 0$. We denote by Sol(II.2) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ that satisfy the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$ To any complex sequence $\left( z _ { k } \right) _ { k \in \mathbb { N } }$, we associate the sequence $\left( Z _ { k } \right) _ { k \in \mathbb { N } }$ of elements of $\mathbb { C } ^ { 2 }$ defined by $$\forall k \in \mathbb { N } , \quad Z _ { k } = \binom { z _ { k } } { z _ { k + 1 } }$$ Prove that the sequence $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ is a solution of (II.2) if and only if the sequence $\left( Z _ { k } \right) _ { k \in \mathbb { N } }$ is a solution of a system (II.3) of the form $$\forall k \in \mathbb { N } , \quad Z _ { k + 1 } = A _ { k } Z _ { k }$$ Specify the matrix $A _ { k } \in \mathcal { M } _ { 2 } ( \mathbb { C } )$.

QII.D.1 Matrices Determinant and Rank Computation View

We denote $Q = A _ { p - 1 } A _ { p - 2 } \cdots A _ { 0 }$. Prove that $\operatorname { det } Q = 1$.

QII.D.2 Matrices Matrix Power Computation and Application View

We denote $Q = A _ { p - 1 } A _ { p - 2 } \cdots A _ { 0 }$. We fix a solution $\left( Z _ { k } \right) _ { k \in \mathbb { N } }$ of (II.3). Prove that, for any natural integer $k$ and any natural integer $r \in \llbracket 1 , p - 1 \rrbracket$, $$\left\{ \begin{array} { l } Z _ { k p } = Q ^ { k } Z _ { 0 } \\ Z _ { k p + r } = A _ { r - 1 } A _ { r - 2 } \cdots A _ { 0 } Q ^ { k } Z _ { 0 } \end{array} \right.$$

QII.E.1 Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix View

We denote $Q = A _ { p - 1 } A _ { p - 2 } \cdots A _ { 0 }$. Prove that (II.2) admits a nonzero periodic solution of period $p$ if and only if 1 is an eigenvalue of $Q$.

QII.E.2 Invariant lines and eigenvalues and vectors Diagonalizability determination or proof View

We denote $Q = A _ { p - 1 } A _ { p - 2 } \cdots A _ { 0 }$. Deduce that (II.2) admits a nonzero periodic solution of period $p$ if and only if $\operatorname { tr } ( Q ) = 2$. Prove that in this case, either all solutions of (II.2) are periodic of period $p$, or (II.2) admits an unbounded solution.
One may prove that there exists a matrix $P \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ and a complex number $\alpha$ such that $Q = P \left( \begin{array} { c c } 1 & \alpha \\ 0 & 1 \end{array} \right) P ^ { - 1 }$ and, in the case where $\alpha \neq 0$, consider the solution of Sol(II.2) whose image by $\Psi$ is the vector $P \binom { 0 } { 1 }$.

QII.E.3 Invariant lines and eigenvalues and vectors Eigenvalue constraints from matrix properties View

We denote $Q = A _ { p - 1 } A _ { p - 2 } \cdots A _ { 0 }$. Show that if $| \operatorname { tr } Q | < 2$, then every solution of (II.2) is bounded.

QIII.A Matrices Linear Transformation and Endomorphism Properties View

Let $n$ and $p$ be two integers greater than or equal to 2. We fix throughout this part a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which we assume to be $p$-periodic, that is such that $\forall k \in \mathbb { N } , A _ { k + p } = A _ { k }$. We denote by $\operatorname { Sol }$ (III.1) the set of sequences $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ of vectors of $\mathbb { C } ^ { n }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } , \quad Y _ { k + 1 } = A _ { k } Y _ { k }$$ Justify that we define a sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ by setting $\left\{ \begin{array} { l } \Phi _ { 0 } = I _ { n } \\ \Phi _ { k + 1 } = A _ { k } \Phi _ { k } \quad \forall k \in \mathbb { N } \end{array} \right.$ and that $\left( Y _ { k } \right) _ { k \in \mathbb { N } } \in \operatorname { Sol }$ (III.1) if and only if $\forall k \in \mathbb { N } , Y _ { k } = \Phi _ { k } Y _ { 0 }$.

QIII.B.1 Matrices Matrix Algebra and Product Properties View

Let $n$ and $p$ be two integers greater than or equal to 2. We fix a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which is $p$-periodic. The sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ is defined by $\Phi _ { 0 } = I _ { n }$ and $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$ for all $k \in \mathbb{N}$. Prove that $\forall k \in \mathbb { N } , \Phi _ { k + p } = \Phi _ { k } \Phi _ { p }$.

QIII.B.2 Invariant lines and eigenvalues and vectors Properties of eigenvalues under matrix operations View

Let $n$ and $p$ be two integers greater than or equal to 2. We fix a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which is $p$-periodic. The sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ is defined by $\Phi _ { 0 } = I _ { n }$ and $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$. The matrix $\Phi _ { p }$ is called the Floquet matrix of equation (III.1) and its complex eigenvalues are called the Floquet multipliers of (III.1). Let $\rho$ be a Floquet multiplier of (III.1).
a) Prove that there exists a nonzero solution $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ of (III.1) satisfying $\forall k \in \mathbb { N } , Y _ { k + p } = \rho Y _ { k }$.
b) Let $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ be such a solution, prove that, if $| \rho | < 1 , \lim _ { k \rightarrow + \infty } \left\| Y _ { k } \right\| _ { \infty } = 0$.

QIII.C Matrices Linear System and Inverse Existence View

Let $n$ and $p$ be two integers greater than or equal to 2. We fix a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which is $p$-periodic. The sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ is defined by $\Phi _ { 0 } = I _ { n }$ and $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$. We denote by $B$ a matrix belonging to $\mathrm { GL } _ { n } ( \mathbb { C } )$ satisfying $B ^ { p } = \Phi _ { p }$. Prove that there exists a unique sequence $\left( P _ { k } \right) _ { k \in \mathbb { N } } \in \left( \mathrm { GL } _ { n } ( \mathbb { C } ) \right) ^ { \mathbb { N } }$, periodic of period $p$, such that $$\forall k \in \mathbb { N } , \quad \Phi _ { k } = P _ { k } B ^ { k }$$

QIII.D.1 Matrices Matrix Norm, Convergence, and Inequality View

Let $n$ and $p$ be two integers greater than or equal to 2. We fix a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which is $p$-periodic. The sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ is defined by $\Phi _ { 0 } = I _ { n }$ and $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$. We denote by $B$ a matrix in $\mathrm { GL } _ { n } ( \mathbb { C } )$ satisfying $B ^ { p } = \Phi _ { p }$, and $\left( P _ { k } \right) _ { k \in \mathbb { N } }$ the unique $p$-periodic sequence in $\left( \mathrm { GL } _ { n } ( \mathbb { C } ) \right) ^ { \mathbb { N } }$ such that $\Phi _ { k } = P _ { k } B ^ { k }$ for all $k \in \mathbb{N}$. Let $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ be a solution of (III.1). Justify the existence of $M = \max _ { k \in \mathbb { N } } \left\| P _ { k } \right\| _ { 0 }$. Show that for all $k \in \mathbb { N } , \left\| \Phi _ { k } \right\| _ { 0 } \leqslant n M \left\| B ^ { k } \right\| _ { 0 }$.

QIII.D.2 Matrices Matrix Norm, Convergence, and Inequality View

Let $n$ and $p$ be two integers greater than or equal to 2. We fix a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which is $p$-periodic. The sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ is defined by $\Phi _ { 0 } = I _ { n }$ and $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$. We denote by $B$ a matrix in $\mathrm { GL } _ { n } ( \mathbb { C } )$ satisfying $B ^ { p } = \Phi _ { p }$, and $\left( P _ { k } \right) _ { k \in \mathbb { N } }$ the unique $p$-periodic sequence such that $\Phi _ { k } = P _ { k } B ^ { k }$. Let $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ be a solution of (III.1).
a) Prove that if $\lim _ { k \rightarrow + \infty } \left\| B ^ { k } \right\| _ { 0 } = 0$, then $\lim _ { k \rightarrow + \infty } \left\| Y _ { k } \right\| _ { \infty } = 0$.
b) Prove that if the sequence $\left( \left\| B ^ { k } \right\| _ { 0 } \right) _ { k \in \mathbb { N } }$ is bounded, then the sequence $\left( \left\| Y _ { k } \right\| _ { \infty } \right) _ { k \in \mathbb { N } }$ is also bounded.

QIII.E.1 Roots of polynomials Multiplicity and derivative analysis of roots View

We still assume that $p$ is an integer greater than or equal to 2. Let $R \in \mathbb { C } [ X ]$ be a polynomial of degree greater than or equal to 1 with simple roots. Prove that the polynomial $R \left( X ^ { p } \right)$ has simple roots if and only if $R ( 0 ) \neq 0$.

QIII.E.2 Invariant lines and eigenvalues and vectors Diagonalizability determination or proof View

We still assume that $p$ is an integer greater than or equal to 2. We denote by $B$ a matrix in $\mathrm { GL } _ { n } ( \mathbb { C } )$ satisfying $B ^ { p } = \Phi _ { p }$. Deduce that $\Phi _ { p }$ is diagonalizable if and only if $B$ is diagonalizable.

QIII.E.3 Invariant lines and eigenvalues and vectors Eigenvalue constraints from matrix properties View

We still assume that $p$ is an integer greater than or equal to 2. We denote by $B$ a matrix in $\mathrm { GL } _ { n } ( \mathbb { C } )$ satisfying $B ^ { p } = \Phi _ { p }$. Suppose that $B$ is diagonalizable and that all its eigenvalues have modulus strictly less than 1. Prove that for every solution $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ of (III.1), $\lim _ { k \rightarrow + \infty } \left\| Y _ { k } \right\| _ { \infty } = 0$.

QIV.A.1 Systems of differential equations View

Let $A$ be a continuous function, periodic of period $T > 0$ and $X$ a function of class $\mathcal { C } ^ { 1 }$ $$A : \left\lvert \, \begin{aligned} & \mathbb { R } \rightarrow \mathcal { M } _ { 2 } ( \mathbb { C } ) \\ & t \mapsto A ( t ) \end{aligned} \quad X \right. : \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto \binom { x _ { 1 } ( t ) } { x _ { 2 } ( t ) } \end{aligned}$$ We are interested in the homogeneous differential system with unknown $X$ $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$. We denote by $U$ and $V$ the two solutions of the differential system (IV.1) satisfying $U \left( t _ { 0 } \right) = \binom { 1 } { 0 }$ and $V \left( t _ { 0 } \right) = \binom { 0 } { 1 }$.
We consider the linear differential system (IV.2) whose solutions are functions of class $\mathcal { C } ^ { 1 }$ with values in $\mathcal { M } _ { 2 } ( \mathbb { C } )$ $$\forall t \in \mathbb { R } , \quad M ^ { \prime } ( t ) = A ( t ) M ( t ) \tag{IV.2}$$ For all $t \in \mathbb { R }$, we set $E ( t ) = [ U ( t ) , V ( t ) ]$. Verify that $E$ is the solution of (IV.2) satisfying $E \left( t _ { 0 } \right) = I _ { 2 }$.

QIV.A.2 Systems of differential equations View

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We consider the linear differential system $$\forall t \in \mathbb { R } , \quad M ^ { \prime } ( t ) = A ( t ) M ( t ) \tag{IV.2}$$ If $M : \left\lvert \, \begin{aligned} & \mathbb { R } \rightarrow \mathcal { M } _ { 2 } ( \mathbb { C } ) \\ & t \mapsto [ F ( t ) , G ( t ) ] \end{aligned} \right.$ is a solution of (IV.2) and $W = \binom { w _ { 1 } } { w _ { 2 } } \in \mathbb { C } ^ { 2 }$, prove that the function $Y : \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto M ( t ) W = w _ { 1 } F ( t ) + w _ { 2 } G ( t ) \end{aligned}$ is a solution of (IV.1).

QIV.B.1 Systems of differential equations View

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$. We denote by $U$ and $V$ the two solutions of (IV.1) satisfying $U \left( t _ { 0 } \right) = \binom { 1 } { 0 }$ and $V \left( t _ { 0 } \right) = \binom { 0 } { 1 }$, and set $E ( t ) = [ U ( t ) , V ( t ) ]$.
Let $t _ { 1 } \in \mathbb { R }$ and $W = \binom { w _ { 1 } } { w _ { 2 } } \in \mathbb { C } ^ { 2 }$. Assume that $E \left( t _ { 1 } \right) W = \binom { 0 } { 0 }$. Show that the function $Y : \left\lvert \, \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto E ( t ) W = w _ { 1 } U ( t ) + w _ { 2 } V ( t ) \end{aligned} \right.$ is zero. Deduce that for all real $t , E ( t )$ is invertible.

QIV.B.2 Systems of differential equations View

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. We consider the linear differential system $$\forall t \in \mathbb { R } , \quad M ^ { \prime } ( t ) = A ( t ) M ( t ) \tag{IV.2}$$ Let $M \in \mathcal { C } ^ { 1 } \left( \mathbb { R } , \mathcal { M } _ { 2 } ( \mathbb { C } ) \right)$ be a solution of system (IV.2). Show that for all real $t , M ( t ) = E ( t ) M \left( t _ { 0 } \right)$.

QIV.B.3 Systems of differential equations View

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. Deduce from the previous question that there exists a unique matrix $B \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ independent of $t$ such that for all real $t , E ( t + T ) = E ( t ) B$.

QIV.C.1 Systems of differential equations View

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. $B \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ is the unique matrix such that $E(t+T) = E(t)B$ for all $t$. The Floquet multipliers of (IV.1) are the eigenvalues of $B$.
Let $\rho \in \mathbb { C }$ be a Floquet multiplier of (IV.1) and $Z \in \mathbb { C } ^ { 2 }$ be an eigenvector of $B$ associated with this eigenvalue. We denote $Y : \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto E ( t ) Z \end{aligned}$.
a) Prove that $\forall t \in \mathbb { R } , Y ( t + T ) = \rho Y ( t )$.
b) Prove that there exists a complex number $\mu$ and a function $S : \left\lvert \, \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto S ( t ) \end{aligned} \right.$ non-zero and $T$-periodic such that $\forall t \in \mathbb { R } , Y ( t ) = \mathrm { e } ^ { \mu t } S ( t )$.

QIV.C.2 Systems of differential equations View

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ $B \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ is the Floquet matrix of (IV.1) and its eigenvalues are the Floquet multipliers. Give a necessary and sufficient condition on the Floquet multipliers for the differential system (IV.1) to admit a non-zero periodic solution of period $T$.

QIV.C.3 Systems of differential equations View

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ $B \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ is the Floquet matrix of (IV.1) and its eigenvalues are the Floquet multipliers. Assume that the matrix $B$ is diagonalizable. Give a necessary and sufficient condition on the Floquet multipliers for the differential system (IV.1) to admit an unbounded solution on $\mathbb { R }$.

QIV.D.1 Systems of differential equations View

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. We set for all $t \in \mathbb { R } , W ( t ) = \operatorname { det } ( E ( t ) )$. Show that for all real $t , W ^ { \prime } ( t ) = \operatorname { tr } ( A ( t ) ) W ( t )$.

QIV.D.2 Systems of differential equations View

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. We set $W ( t ) = \operatorname { det } ( E ( t ) )$ and denote $\rho _ { 1 }$ and $\rho _ { 2 }$ the Floquet multipliers of (IV.1). Deduce that $\rho _ { 1 } \rho _ { 2 } = \exp \left( \int _ { 0 } ^ { T } \operatorname { tr } ( A ( s ) ) \mathrm { d } s \right)$.

QV.A Matrices Matrix Power Computation and Application View

We fix a natural number $p$ greater than or equal to 2. We recall the following result: for all matrices $A _ { 1 }$ and $A _ { 2 }$ in $\mathcal { M } _ { n } ( \mathbb { C } )$, all matrices $X _ { 1 }$ and $X _ { 2 }$ in $\mathcal { M } _ { n , 1 } ( \mathbb { C } )$ and all complex numbers $\lambda _ { 1 }$ and $\lambda _ { 2 }$: $$\left( \begin{array} { c c } A _ { 1 } & X _ { 1 } \\ 0 _ { 1 , n } & \lambda _ { 1 } \end{array} \right) \left( \begin{array} { c c } A _ { 2 } & X _ { 2 } \\ 0 _ { 1 , n } & \lambda _ { 2 } \end{array} \right) = \left( \begin{array} { c c } A _ { 1 } A _ { 2 } & A _ { 1 } X _ { 2 } + \lambda _ { 2 } X _ { 1 } \\ 0 _ { 1 , n } & \lambda _ { 1 } \lambda _ { 2 } \end{array} \right)$$ Let $A \in \mathcal { M } _ { n } ( \mathbb { C } ) , X \in \mathcal { M } _ { n , 1 } ( \mathbb { C } )$ and $\lambda \in \mathbb { C }$. Prove that, for all integer $k \geqslant 1$ we have: $$\left( \begin{array} { c c } A & X \\ 0 _ { 1 , n } & \lambda \end{array} \right) ^ { k } = \left( \begin{array} { c c } A ^ { k } & X _ { k } \\ 0 _ { 1 , n } & \lambda ^ { k } \end{array} \right)$$ where $X _ { k } = \left( \sum _ { j = 0 } ^ { k - 1 } \lambda ^ { k - 1 - j } A ^ { j } \right) X$.

QV.B.1 Complex Numbers Arithmetic Roots of Unity and Cyclotomic Expressions View

We fix a natural number $p$ greater than or equal to 2. We denote $\mathcal { V } _ { p } = \left\{ \mathrm { e } ^ { \frac { 2 \mathrm { i } k \pi } { p } } ; k \in \llbracket 1 , p - 1 \rrbracket \right\}$, the set of $p$-th roots of unity different from 1. Let $a$ and $\lambda$ be non-zero complex numbers. Assume that $\frac { a } { \lambda } \notin \mathcal { V } _ { p }$, which means that either $a = \lambda$ or $\frac { a ^ { p } } { \lambda ^ { p } } \neq 1$. Prove that the complex number $\sum _ { j = 0 } ^ { p - 1 } \lambda ^ { p - 1 - j } a ^ { j }$ is non-zero.

QV.B.2 Matrices Determinant and Rank Computation View

We fix a natural number $p$ greater than or equal to 2. We denote $\mathcal { V } _ { p } = \left\{ \mathrm { e } ^ { \frac { 2 \mathrm { i } k \pi } { p } } ; k \in \llbracket 1 , p - 1 \rrbracket \right\}$, the set of $p$-th roots of unity different from 1. Let $A = \left( a _ { i , j } \right)$ be a matrix in $\mathcal { M } _ { n } ( \mathbb { C } )$ that is upper triangular and invertible. Let $\lambda$ be a non-zero complex number. Assume that, for all $i \in \llbracket 1 , n \rrbracket , \frac { a _ { i , i } } { \lambda } \notin \mathcal { V } _ { p }$. Prove that the matrix $\sum _ { j = 0 } ^ { p - 1 } \lambda ^ { p - 1 - j } A ^ { j }$ is invertible.

QV.B.3 Proof by induction Prove existence or uniqueness of an object by induction View

We fix a natural number $p$ greater than or equal to 2. Show that every upper triangular and invertible matrix admits at least one upper triangular $p$-th root.
One may prove by induction on $n \geqslant 1$ the following property: $$\forall B \in \mathcal { T } _ { n } ( \mathbb { C } ) \cap \mathrm { GL } _ { n } ( \mathbb { C } ) , \quad \exists A \in \mathcal { T } _ { n } ( \mathbb { C } ) \quad \text { such that } \left\{ \begin{array} { l } A ^ { p } = B \\ \forall ( i , j ) \in \llbracket 1 , n \rrbracket ^ { 2 } , \frac { a _ { i , i } } { a _ { j , j } } \notin \mathcal { V } _ { p } \end{array} \right.$$

QV.B.4 Matrices Matrix Decomposition and Factorization View

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).

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