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Given a matrix $A$ of size $n\times n$, which need not be symmetric, $A^2 + (A^2)^t$ (where $t$ denotes transpose) is certainly symmetric. Is it positive semidefinite?

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The answer is no. Example: $$ A = \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right) $$

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