# Constraint on M to keep M^T * A * M positive semidefinite?

1. Dec 8, 2015

### perplexabot

Hey all! Let me get right to it!
It is given that $$A \succeq 0$$
I need the following to hold for $M$: $$M^TAM\succeq 0$$

What are the constraints or conditions on $M$ for $M^TAM\succeq 0$ to hold?

Anything would help at this point... I am open to discussion.

Note: It may be worth mentioning that $A$ is a given matrix where as $M$ is variable.

Thank you for reading : )

Last edited: Dec 8, 2015
2. Dec 8, 2015

### micromass

It is always true. For any $M$.

3. Dec 8, 2015

### perplexabot

Interesting, I think I see why that is. $xM^TAMx \succeq 0 => y^TAy \succeq 0$

Thanks for the quick answer. I may have a follow up question later :P