grandes-ecoles 2017 QII.D.2

grandes-ecoles · France · centrale-maths2__psi Matrices Matrix Power Computation and Application
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 $r \in \llbracket 1 , p - 1 \rrbracket$, $$\left\{ \begin{array} { l } Z _ { k p } = Q ^ { k } Z _ { 0 } \\ Z _ { k p + r } = A _ { r - 1 } A _ { r - 2 } \cdots A _ { 0 } Q ^ { k } Z _ { 0 } \end{array} \right.$$ 
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 $r \in \llbracket 1 , p - 1 \rrbracket$,
$$\left\{ \begin{array} { l } Z _ { k p } = Q ^ { k } Z _ { 0 } \\ Z _ { k p + r } = A _ { r - 1 } A _ { r - 2 } \cdots A _ { 0 } Q ^ { k } Z _ { 0 } \end{array} \right.$$
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