Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question

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. $A = 0, B = - I_n, C = I_n$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.