Suppose $X$ and $Y$ are real matrices such that $A=X+iY$ is Hermitian and positive definite. Does invertability of $A$ imply invertability of $X$? Are all Hermitian positive definite matrices invertible?
2026-04-08 07:31:51.1775633511
Is the real part of a positive definite Hermitian matrix invertible?
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The answer is yes. In particular, it suffices to note that $$ X = \frac 12A + \frac 12 \overline{A} $$ is a sum of positive definite matrices.