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$W$ is a positive definite matrix.

$C=\pmatrix{-H & -A^T\\\ A & 0}$, where $H>0$ and $A$ is full rank. (it can be shown that all eigenvalues of $C$ have negative real part.) We know that (can be shown) that all the eigenvalues of $WC$ has negative real part. What is the relationship between the eigenvalues of $WC$ and $C$? Can we show that if $W>I$ results in real part of eigenvalues of $CW$ being more negative that $C$'s?

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