Spectrum of SPD block matrix

Question

Let $M=\left[\begin{array}{cc}A & B \\B^T & C \end{array}\right]\in\mathbb{R}^{(n+m)\times (n+m)}$ be a symmetric positive definite matrix, where $n\geq m$, $A\in\mathbb{R}^{n\times n}$, $C\in\mathbb{R}^{m\times m}$ are symmetric positive definite matrices and $B\in\mathbb{R}^{n\times m}$ be a full rank matrix, i.e., $\text{rank}(B)=m$. What is the spectrum of $M$ in terms of its blocks? I am particularly interested in having a lower bound on the minimum eigenvalue of $M$. I would highly appreciate any suggestions or references.

Answer 1

$X:=\left[\begin{array}{cc}A & \mathbf 0 \\\mathbf 0 & C \end{array}\right]=M-\left[\begin{array}{cc}\mathbf 0 & B\\ B^T &\mathbf 0 & \end{array}\right]$

$\lambda_{min}\Big(M\Big)$
$= \lambda_{min} \left( X + \left[\begin{array}{cc}\mathbf 0 & B\\ B^T &\mathbf 0 & \end{array}\right]\right)$
$\geq \lambda_{min} \left( X\right) + \lambda_{min} \left(\left[\begin{array}{cc}\mathbf 0 & B\\ B^T &\mathbf 0 & \end{array}\right]\right)$
$= \lambda_{min} \left( X\right) + -\lambda_{max} \left(BB^T\right)^\frac{1}{2}$
$= \lambda_{min} \left( X\right) -\sigma_{max} \left(B\right)$
$= \min\Big(\lambda_{min}( A),\lambda_{min}(C)\Big) -\sigma_{max} \left(B\right)$
This bound is sharp --- if you play around with projection matrices, it's easy to construct cases where this is met with equality.

Something else of interest
$M \succeq X$ i.e. $M$ majorizes $X$ and