Riv. Mat. Univ. Parma, Vol. 9, No. 2, 2018

Giuseppe Buffoni [a]

On the structure of matrices with positive inverse

Pages: 191-226
Accepted in revised form: 24 January 2019
Mathematics Subject Classification (2010): 15B48, 65F30.
Keywords: Monotone matrices, nonnegative and nonpositive perturbations, monotonicity properties of the inverse, preserving monotonicity.
[a]: CNR - IMATI, Via Bassini 15, 20133 Milano, Italy

Abstract: This paper focuses on how monotone+ matrices, i.e., real nonsingular matrices with positive inverse, can either be perturbed or decomposed in such a way that the inverse-positivity is preserved and proved. Let a real matrix $$A$$ be split into its components: diagonal entries $$D$$, nonpositive $$-B$$ and nonnegative $$C$$ off-diagonal entries: $$A=D-B+C$$. Monotone+ matrices with only two components and their perturbations are identified by investigating the properties of the splittings $$D-B$$, $$D+C$$ and $$D-B+C$$. Monotone+ matrices characterized by three components are identified by means of more involved decompositions of $$A$$ or suitable transformations of $$A$$, preserving the inverse-positivity, that emphasize the basic properties leading to inverse positivity. Special complex monotone+ matrices are described. The analysis is strongly based on some monotonicity properties of nonpositive and nonnegative perturbations of a monotone+ matrix preserving the inverse-positivity. The results are illustrated by numerical examples.

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