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

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The discussion centers on the mathematical condition for a variable matrix M such that the expression \(M^T A M \succeq 0\) holds true, given that matrix A is positive semidefinite (A ≽ 0). It is established that this condition is satisfied for any matrix M, as the positive semidefiniteness of A guarantees that \(x^T M^T A M x \succeq 0\) for all vectors x. The participants confirm that the condition is universally valid without additional constraints on M.

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

What are the constraints or conditions on [itex]M[/itex] for [itex]M^TAM\succeq 0[/itex] to hold?

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

Note: It may be worth mentioning that [itex]A[/itex] is a given matrix where as [itex]M[/itex] is variable.

Thank you for reading : )
 
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It is always true. For any ##M##.
 
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micromass said:
It is always true. For any ##M##.
Interesting, I think I see why that is. [itex]xM^TAMx \succeq 0 => y^TAy \succeq 0[/itex]

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

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