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If $A-B$ and $C$ are both positive semi-definite, is $ACA-BCB$ also positive semi-definite?

Welcome proof, reference, or counter example.

Thanks a lot in advance.

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No, it's not even true for scalars (aka 1×1 matrices). Take: $$A=0,B=−1,C=1$$ Then $A−B=1$ is positive semi-definite (even positive definite), so is C, but $ACA−BCB=−1$ is not.

It extends in every dimension by setting eg. $A = 0, B = - I_n, C = I_n$.

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