grandes-ecoles 2013 QIV.A.1

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

Let $A \in M_p(\mathbb{C})$ and $k \in \mathbb{N}^*$ such that $A^k = 0$ and $A^{k-1} \neq 0$ (we say that $A$ is nilpotent of order $k$).
Show that, for every integer $j$ such that $1 \leqslant j \leqslant k$, $\operatorname{Ker} A^{j-1}$ is strictly included in $\operatorname{Ker} A^j$.

Let $A \in M_p(\mathbb{C})$ and $k \in \mathbb{N}^*$ such that $A^k = 0$ and $A^{k-1} \neq 0$ (we say that $A$ is nilpotent of order $k$).

Show that, for every integer $j$ such that $1 \leqslant j \leqslant k$, $\operatorname{Ker} A^{j-1}$ is strictly included in $\operatorname{Ker} A^j$.

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