grandes-ecoles 2016 QV.B.1

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

For the positive matrix $A$ of $\mathcal{M}_n(\mathbb{R})$, show that the following conditions are equivalent:

the matrix $A$ is irreducible;
the matrix $B = I_n + A + A^2 + \cdots + A^{n-1}$ is strictly positive;
the matrix $C = (I_n + A)^{n-1}$ is strictly positive.

For the positive matrix $A$ of $\mathcal{M}_n(\mathbb{R})$, show that the following conditions are equivalent:
\begin{itemize}
  \item the matrix $A$ is irreducible;
  \item the matrix $B = I_n + A + A^2 + \cdots + A^{n-1}$ is strictly positive;
  \item the matrix $C = (I_n + A)^{n-1}$ is strictly positive.
\end{itemize}

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