Given matrix $B$ is positive definite, and $B\geq \underline{b}I$, $AA^T$ is positive semi-definite. Can the following inequality hold? $$ABA^T\geq \underline{b}AA^T.$$
2026-03-30 11:59:13.1774871953
Given a positive definite matrix $B$ , $B\geq \underline{b}I$, $AA^T$ is positive semi-definite. Can $ABA^T\geq b AA^T$?
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$$ x^TABA^Tx = (A^Tx)^TB(A^Tx)\ge b(A^Tx)^T(A^Tx) = x^T(bAA^T)x. $$