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Is it true that for matrices of any sizes (for which the following makes sense), that if $P$ is a symmetric PSD projection matrix,

$$\|APB\|_2 \leq \|AB\|_2?$$

where $\|\cdot\|$ is the operator norm?

Are there any conditions one can put on $A$ and $B$ that make this true?

bringingdownthegauss
  • 530

1 Answers1

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No. Counterexample: $APB=\pmatrix{1&1\\ 0&0}\pmatrix{1\\ &0}\pmatrix{1&0\\ -1&0}\ne0$, but $AB=0$.

user1551
  • 139,064