grandes-ecoles 2021 Q2

grandes-ecoles · France · centrale-maths1__official Matrices Diagonalizability and Similarity

We denote $D_{A} = \operatorname{diag}(\lambda_{1}(A), \ldots, \lambda_{n}(A))$ and $D_{B} = \operatorname{diag}(\lambda_{1}(B), \ldots, \lambda_{n}(B))$. Show that there exists an orthogonal matrix $P = (p_{i,j})_{1 \leqslant i,j \leqslant n}$ such that $\|A - B\|_{F}^{2} = \|D_{A}P - PD_{B}\|_{F}^{2}$.

We denote $D_{A} = \operatorname{diag}(\lambda_{1}(A), \ldots, \lambda_{n}(A))$ and $D_{B} = \operatorname{diag}(\lambda_{1}(B), \ldots, \lambda_{n}(B))$. Show that there exists an orthogonal matrix $P = (p_{i,j})_{1 \leqslant i,j \leqslant n}$ such that $\|A - B\|_{F}^{2} = \|D_{A}P - PD_{B}\|_{F}^{2}$.

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