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 $B$?