A necessary condition for certain functions to preserve positive semi-definiteness on partitioned matrices

Abstract

If $f$ is a symmetric complex-valued function on the $m$-fold Cartesian product of the set of non-negative reals and $A$ is a positive semi-definite $m\times m$ matrix with eigenvalues ${\lambda }_j$, we set $f(A):=f({\lambda }_1,\dots ,{\lambda }_m)$. It is shown that if $[f(A_{\alpha \beta })]$ is positive semi-definite whenever $[A_{\alpha \beta }]$ is a positive semi-definite matrix with positive semi-definite entries $A_{\alpha \beta }$, then $f$ has a power series expansion with positive coefficients.