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Let $A = \mathrm{Diag}(\lambda_1, \dots, \lambda_n)$ where $\lambda_1 \le \lambda_2 \dots \le \lambda_n$. Let $P = I - ww^T$ be a projection operator on an arbitrary $n$-dimensional hyperplane. Let $B = PAP$ have eigenvalues $\mu_1 \le \mu_2 \le \dots \mu_{n - 1}$ and (unit norm) eigenvectors $v_1, \dots, v_{n - 1}$. From Cauchy's interlacing theorem it follows that $\lambda_1 \le \mu_1 \le \lambda_2 \le \dots \mu_{n-1} \le \lambda_n$.

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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  • $\begingroup$ Thinking in 3D, the projection of relationship $AV=\lambda V$ gives rise to $BV'=\mu V'$ with $\mu=\lambda$... which is a particular case of the interlacing property. $\endgroup$ Commented Jan 19, 2020 at 9:12

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