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$ \newcommand\tr{\text{tr}} \newcommand\mean[1]{\left\langle{#1}\right\rangle}$ Let $ B\in\mathbb R^{n\times n}$ with its eigenvalues restricted to the left half plane, and anti-symmetric matrix $ T\in\mathbb R^{n\times n}$ satisfy $ $ B^\top-TB^\top=B+BT.$ $ Prove that $ \tr(TB)\leq0$ . What are necessary and sufficient conditions on $ B$ such that $ \tr(TB)=0$ . (e.g. it is sufficient for $ B$Read more