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eigenvalues of matrices or operators

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Relationship between eigenvectors of projected and original matrix

Let $B = PAP$ have eigenvalues $\mu_1 \le \mu_2 \le \dots \mu_{n - 1}$ and (unit norm) eigenvectors $v_1, \dots, v_{n - 1}$. … Is there anything that can be said about the relationship between inner products $\langle e_i, v_j \rangle$ depending on the eigenvalues of $A$ and $B$? …
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