grandes-ecoles 2025 Q5

grandes-ecoles · France · x-ens-maths__pc Matrices Diagonalizability and Similarity

Let $K \in \mathcal{M}_n(\mathbb{R})$ be a matrix of rank 1, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$ be such that $K = \mathbf{u v}^T$.
(a) Show that $\operatorname{Tr}(K) = \langle \mathbf{v}, \mathbf{u} \rangle$.
(b) Show that $K^2 = \operatorname{Tr}(K) K$.
(c) Deduce that $K$ is diagonalizable if and only if $\operatorname{Tr}(K) \neq 0$.

Let $K \in \mathcal{M}_n(\mathbb{R})$ be a matrix of rank 1, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$ be such that $K = \mathbf{u v}^T$.\\
(a) Show that $\operatorname{Tr}(K) = \langle \mathbf{v}, \mathbf{u} \rangle$.\\
(b) Show that $K^2 = \operatorname{Tr}(K) K$.\\
(c) Deduce that $K$ is diagonalizable if and only if $\operatorname{Tr}(K) \neq 0$.

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