grandes-ecoles 2013 QV.B.2

grandes-ecoles · France · centrale-maths2__psi Matrices Linear Transformation and Endomorphism Properties

Let $A \in M_p(\mathbb{C})$ and $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ as defined in V.A.4. Let $P$ be a non-zero annihilating polynomial of the matrix $B$.
a) Show that the degree of $P$ is $\geqslant p$.
b) Deduce that the family $\left\{B^i,\ 0 \leqslant i \leqslant p-1\right\}$ is free.

Let $A \in M_p(\mathbb{C})$ and $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ as defined in V.A.4. Let $P$ be a non-zero annihilating polynomial of the matrix $B$.

a) Show that the degree of $P$ is $\geqslant p$.

b) Deduce that the family $\left\{B^i,\ 0 \leqslant i \leqslant p-1\right\}$ is free.

Paper Questions

QI.A QI.B QII.A.1 QII.A.2 QII.A.3 QII.B.1 QII.B.2 QII.B.3 QII.B.4 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2 QIII.B.3 QIII.C.1 QIII.C.2 QIV.A.1 QIV.A.2 QIV.B QIV.C QIV.D QIV.E QIV.F QV.A.1 QV.A.2 QV.A.3 QV.A.4 QV.B.1 QV.B.2 QV.B.3 QV.B.4