Is the set of positive definite matrices with trace one an open subset of hermitian matrices?
I know the set of positive definite matrices is open, but I don't know how to prove that the trace one condition won't affect openness.
2026-03-25 22:05:14.1774476314
Is the set of positive definite matrices with trace one an open subset of hermitian matrices?
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No, it isn't open. If $A$ is a positive definite matrix with trace $1$, then $A+tI$ is also positive definite for every $t>0$.