grandes-ecoles 2021 Q2

grandes-ecoles · France · centrale-maths1__mp Matrices Diagonalizability and Similarity

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

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

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