Riv. Mat. Univ. Parma, Vol. 9, No. 2, 2018
Giuseppe Buffoni [a]
On the structure of matrices with positive inverse
Pages: 191-226
Received: 5 July 2018
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.
Author address:
[a]: CNR - IMATI, Via Bassini 15, 20133 Milano, Italy
Full Text (PDF)
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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