grandes-ecoles 2017 QIII.E.3

grandes-ecoles · France · centrale-maths2__psi Invariant lines and eigenvalues and vectors Eigenvalue constraints from matrix properties
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), $\lim _ { k \rightarrow + \infty } \left\| Y _ { k } \right\| _ { \infty } = 0$. 
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), $\lim _ { k \rightarrow + \infty } \left\| Y _ { k } \right\| _ { \infty } = 0$.
Paper Questions
QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.A.5 QII.B.1 QII.B.2 QII.B.3 QII.C QII.D.1 QII.D.2 QII.E.1 QII.E.2 QII.E.3 QIII.A QIII.B.1 QIII.B.2 QIII.C QIII.D.1 QIII.D.2 QIII.E.1 QIII.E.2 QIII.E.3 QIV.A.1 QIV.A.2 QIV.B.1 QIV.B.2 QIV.B.3 QIV.C.1 QIV.C.2 QIV.C.3 QIV.D.1 QIV.D.2 QV.A QV.B.1 QV.B.2 QV.B.3 QV.B.4