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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}

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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).

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  • $\begingroup$ what is inertia? $\endgroup$ Dec 4, 2019 at 1:32
  • $\begingroup$ Please refer to the hyperlink I added to the updated answer. It means the ordered set of the count of positive, negative and zero eigenvalues of a matrix. $\endgroup$ Dec 4, 2019 at 2:56
  • $\begingroup$ I don't understand what is the addition of "inertias" here? Are the sets added component-wise? $\endgroup$
    – a06e
    Apr 27, 2020 at 22:01

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