Conditions on negative definiteness

[p4wp4w]

Hello, I want to know if there exist any result in literature that answers my question:
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.

 

[perplexabot]

Have you tried looking at Schur complements? I'm not sure it would help, but maybe.

 

[p4wp4w]

Have you tried looking at Schur complements? I'm not sure it would help, but maybe.

No, it won't help really. The matrix inequality is already linear (LMI) and Schur complement in that sense will just cause a second problem that the off-diagonal terms will cause nonlinearty (later $R$ should be found by interior point method). I am expecting the answer to be in the form of a second LMI on $R$ but the problem is that a lot of simplifications or even assumptions can not be done since $R$ is not square.

 

[perplexabot]

No, it won't help really. The matrix inequality is already linear (LMI) and Schur complement in that sense will just cause a second problem that the off-diagonal terms will cause nonlinearty (later $R$ should be found by interior point method). I am expecting the answer to be in the form of a second LMI on $R$ but the problem is that a lot of simplifications or even assumptions can not be done since $R$ is not square.

Hmmm. Maybe you are right, Schur may not be of use. I was just throwing things out there.

Wait, you say in your last post that $R$ is not square but in your original post you say it is positive definite?! Which one is it?
If $R$ is positive definite and if you assume the first argument (1) is true, then one condition on $R$ that results in argument (2) being satisfied is if $R$ is simply the Identity matrix : P

 

[p4wp4w]

Hmmm. Maybe you are right, Schur may not be of use. I was just throwing things out there.

Wait, you say in your last post that $R$ is not square but in your original post you say it is positive definite?! Which one is it?
If $R$ is positive definite and if you assume the first argument (1) is true, then one condition on $R$ that results in argument (2) being satisfied is if $R$ is simply the Identity matrix : P

That's a nasty mistake that I made; I think deep down, I was looking for something like SPDness of $R$ to simplify things but $R$ is not square in general.

 

[perplexabot]

That's a nasty mistake that I made; I think deep down, I was looking for something like SPDness of $R$ to simplify things but $R$ is not square in general.

Can you assume $A$ or $D$ to be positive definite?