Let $A$ be a positive definite matrix and $B$ be a positive semi-definite matrix. Then $AB$ is diagonalizable.
I want to see if this is true or false.
2026-03-25 07:45:29.1774424729
Let $A$ be a positive definite matrix and $B$ be a positive semi-definite matrix. Then $AB$ is diagonalizable.
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It's wrong. $A=\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}$ and $B=\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}$.