If Matrix $A$ is positive definite. Does it hold for every $A$ that $-A$ is negative definite? Does the same hold for positive semidefinite $A$ too, that $-A$ is negative semidefinite?
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2026-03-29 08:21:06.1774772466
Is a negated positive (semi-)definite matrix always negative (semi-)definite and vice versa?
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Yes. Because (for instance) $x'(-A)x = -x'Ax$, so any vector $x$ that is a witness for or against the positive definiteness of $A$ is also a witness for or against the negative definiteness of $-A$, and so on.