Functions operating on positive-definite symmetric matrices
For a matrix \(A=(a_{ij})\in\mathbb{R}^{n\times n}\) with nonnegative entries one defines \(f(A):=(f(a_{ij}))\), where \(f:\mathbb{R}^ +_ 0\to\mathbb{R}^ +_ 0\) (if \(f\) is a polynomial, this means that \(f(A)\) is evaluated using the Schur product). Denoting with \({\mathcal S}^ +_ n\) the set of all nonnegative-definite symmetric matrices with nonnegative entries one is interested in functions \(f\) such that \(f({\mathcal S}^ +_ n)\subseteq{\mathcal S}^ +_ n\). In the case where \(f(t)=\sum^ N_{k=1}c_ it^{\alpha_ i}\), \(0<\alpha_ 1<\cdots<\alpha_ N\), the authors give some necessary conditions on \(f\) such that this inclusion holds: They prove that \(c_ i>0\) must hold for \(i\leq n\) and for \(i\geq N-n+1\) if \(f(A)\in{\mathcal S}^ +_ n\) for each rank one matrix \(A\in{\mathcal S}^ +_ n\). An example with \(N=5\), \(n=2\) shows that this condition (in terms of rank one matrices) cannot be improved very much.
scientific article; zbMATH DE number 150707