If the determinant of the hermitian part of a matrix $A$ is positive, can I say that $\det A$ is positive too? If so, how can I prove it?
2026-04-09 08:01:45.1775721705
Positive determinant of matrix
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No, a counterexample is $$A = \pmatrix{-2 & -4 & -2 \\ 2 & 2 & 3 \\ 1 & -4 & -5}$$ whose determinant is $-36$ but whose symmetric part, $\frac{1}{2}(A+A^T),$ has determinant $\frac{49}{2}$.