Suppose that $M$ is a positive definite matrix with entries within $[-1,1]$, and let $N$ be a matrix where $N_{ij} = \sin^{-1}M_{ij}$. How do I show that $N$ is also positive definite?
2026-03-25 23:38:15.1774481895
Positive Definiteness of Arcsin of a Positve Definite Matrix
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This follows from the facts that the element-wise product of p.s.d. matrices is p.s.d, that the sum of p.s.d. matrices is p.s.d., and that the Taylor series for $\arcsin$ has non-negative terms.