For a symmetric invertible block matrix such as below, is there a relation between the eigenvalues of $M$ and that of the Schur complements and the matrices in the diagonal?
\begin{align} M = \begin{bmatrix}A & B \\ B^T & C\end{bmatrix} \end{align}
2025-01-12 23:48:00.1736725680
Eigenvalues of the Schur complement
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Yes, from the Haynsworth inertia additivity formula
$$ \text{In}(M)=\text{In}(C)+\text{In}(\underbrace{A-BC^{-1}B^T}_{M~\setminus~ C})=\text{In}(A)+\text{In}(\underbrace{C-B^TA^{-1}B}_{M~\setminus~ A}) $$
where $\text{In}()$ denotes the inertia (ordered set of the count of positive, negative and zero eigenvalues of a matrix).