As the title suggests, I am wondering whether we can define a function $f$ over the set of positive definite real matrices $f: \mathbb S_{+}^n \to \mathbb R$ such that $f$ has the property: $f(X) \ge f(Y) \implies X \succeq Y$?
2026-03-25 11:16:33.1774437393
Is it possible to define a scalar valued function $f$ over positive definite matrices, such that $f(X) \le f(Y)$ implies $X \preceq Y$?
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