grandes-ecoles 2016 QV.C.2

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

In this question, $A$ is a given irreducible matrix.
Suppose that: $\exists i \in \llbracket 1, n \rrbracket, a_{i,i} > 0$. Show that $A$ is primitive.
For all $j$ and $k$ in $\llbracket 1, n \rrbracket$, one can show that there exists in $A$ a path from $j$ to $k$ passing through $i$, and consider the maximum $m$ of the lengths of the paths thus obtained. One will prove that $A^m > 0$.

In this question, $A$ is a given irreducible matrix.

Suppose that: $\exists i \in \llbracket 1, n \rrbracket, a_{i,i} > 0$. Show that $A$ is primitive.

For all $j$ and $k$ in $\llbracket 1, n \rrbracket$, one can show that there exists in $A$ a path from $j$ to $k$ passing through $i$, and consider the maximum $m$ of the lengths of the paths thus obtained. One will prove that $A^m > 0$.

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