Functions preserving nonnegativity of matrices

Details Functions preserving nonnegativity of matrices Functions preserving nonnegativity of matrices

2005-11-13

arxiv.org/abs/math/0511325v2

SIAM J. Matrix Anal. Appl. 30 (2008) no.1, 84-101

Mathematics

Rings and Algebras

20 pages; expanded and corrected to reflect referees' remarks; to appear in SIAM J. Matrix Anal. Appl

Scientific paper

10.1137/050645075

The main goal of this work is to determine which entire functions preserve nonnegativity of matrices of a fixed order $n$ -- i.e., to characterize entire functions $f$ with the property that $f(A)$ is entrywise nonnegative for every entrywise nonnegative matrix $A$ of size $n\times n$. Towards this goal, we present a complete characterization of functions preserving nonnegativity of (block) upper-triangular matrices and those preserving nonnegativity of circulant matrices. We also derive necessary conditions and sufficient conditions for entire functions that preserve nonnegativity of symmetric matrices. We also show that some of these latter conditions characterize the even or odd functions that preserve nonnegativity of symmetric matrices.

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