Let $A$ be a strictly positive symmetric matrix in $\mathbb{R}^{n \times n}$ and $P$ be an orthogonal projection onto a subspace $U$ of dimension $m \leq n$.
Is there any way to relate the eigenvalues (and/or eigenvectors) of the projected matrix $PAP$ to the matrix $A$?
My end goal is to find a bound (if possible) on $\|(PAP+ \lambda)^{-1/2}(A+\lambda)^{1/2}\|$ where $\|\cdot\|$ is the operator norm and $\lambda > 0$.
Any pointer/reference welcome.
Without loss of generality we assume that the subspace is $R^m$. As a consequence, by blocks: $$PAP=\left[\begin{array}{cc}I_m&0\\0&0\end{array}\right] \left[\begin{array}{cc}A_1&A_{12}\\A_{21}&A_2\end{array}\right] \left[\begin{array}{cc}I_m&0\\0&0\end{array}\right] = \left[\begin{array}{cc}A_1&0\\0&0\end{array}\right].$$ Suppose that $m=n-1$ and that the eigenvalues of $A$ and $A_1$ are respectively $$\lambda_1\leq\ldots\leq\lambda_n,\ \mu_1\leq\ldots\leq \mu_{n-1}.$$ Then $$\lambda_1\leq \mu_1\leq \lambda_2\leq \mu_2\leq\ldots\leq \mu_{n-1}\leq\lambda_n.$$ The proof is a bit long to be given here but is found in the good books of linear algebra.
Well, if $m$ is $n-2$ or smaller, you have to iterate this result, and the location of the eigenvalues of $A_1$ will be less and less precise when $m$ decreases.