Hello, I want to know if there exist any result in literature that answers my question:(adsbygoogle = window.adsbygoogle || []).push({});

Under which conditions on the real valued matrix ## R ## (symmetric positive definite), the first argument results in and guarantees the second one:

1) for real valued matrices ##A, B, C,## and ## D ## with appropriate dimensions and ## A ## and ## D ## being symmetric:

##X=

\begin{pmatrix}

A & B+RC\\

B^T+C^TR & D\\

\end{pmatrix} < 0##

2)

##

Y

=

\begin{pmatrix}

A & B+C\\

B^T+C^T & D\\

\end{pmatrix} < 0##

Congruence transformation doesn't help since it will affect the diagonal elements as well.

Thank you all in advance.

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# I Conditions on negative definiteness

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