grandes-ecoles 2016 QVI.A

grandes-ecoles · France · centrale-maths1__mp Matrices Matrix Power Computation and Application

Let $A$ be an imprimitiv matrix with coefficient of imprimitivity $p \geqslant 2$.
For any integer $m$ that is not a multiple of $p$, show that the diagonal of $A^m$ is identically zero. One can be interested in the trace of $A^m$.
Deduce that the result of question IV.B.3 no longer holds if $A$ is imprimitiv.

Let $A$ be an imprimitiv matrix with coefficient of imprimitivity $p \geqslant 2$.

For any integer $m$ that is not a multiple of $p$, show that the diagonal of $A^m$ is identically zero. One can be interested in the trace of $A^m$.

Deduce that the result of question IV.B.3 no longer holds if $A$ is imprimitiv.

Paper Questions

QI.A.1 QI.A.2 QI.B.1 QI.B.2 QII.A QII.B QII.C QIII.A QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIII.B.5 QIII.C.1 QIII.C.2 QIII.C.3 QIII.D.1 QIII.D.2 QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.B.1 QIV.B.2 QIV.B.3 QIV.C QV.A.1 QV.A.2 QV.A.3 QV.A.4 QV.A.5 QV.B.1 QV.B.2 QV.C.1 QV.C.2 QVI.A QVI.B.1 QVI.B.2 QVI.B.3 QVI.C.1 QVI.C.2 QVI.D