Analytic functions ofM-matrices and generalizations

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Abstract
We introduce the notion of positivity cone K of matrices in and with such a K we associate sets Z and M. For suitable choices of K the set M consists of the classical (non-singular) M-matriccs or of the positive definite (Hermitian) matrices. If AϵM and 1⩽p⩽3 we prove that there is a unique BϵM for which Bp =A. If P>3, this uniqueness theorem is false for general M and we prove a weaker result. We extend the result that for a Z-matrix A we have A −1⩾0 if and only if A is an M-matrix. Under an additional hypothesis on the positivity cone, we exhibit a class of entire functions f(z) such that for A ϵ Z we have A ϵ M if and only if there is a B ϵ K for which f(B) = A −1.

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