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.

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