grandes-ecoles 2025 Q16

grandes-ecoles · France · x-ens-maths__pc Matrices Eigenvalue and Characteristic Polynomial Analysis

Let $\lambda$ be an eigenvalue of $A$ with multiplicity $m \geqslant 2$. We set $E = \operatorname{Ker}\left(A - \lambda \mathbb{I}_n\right)$.
(a) Show that $\operatorname{dim}\left(E \cap \{\mathbf{u}\}^\perp\right) \geqslant m - 1$.
(b) Deduce that $\lambda$ is an eigenvalue of $B$ with multiplicity at least $m - 1$.

Let $\lambda$ be an eigenvalue of $A$ with multiplicity $m \geqslant 2$. We set $E = \operatorname{Ker}\left(A - \lambda \mathbb{I}_n\right)$.\\
(a) Show that $\operatorname{dim}\left(E \cap \{\mathbf{u}\}^\perp\right) \geqslant m - 1$.\\
(b) Deduce that $\lambda$ is an eigenvalue of $B$ with multiplicity at least $m - 1$.

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