Riv. Mat. Univ. Parma, Vol. 9, No. 2, 2018
Giuseppe Buffoni ^{[a]}
On the structure of matrices with positive inverse
Pages: 191226
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
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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 inversepositivity is preserved and proved.
Let a real matrix \(A\) be split into its components: diagonal entries \(D\), nonpositive
\(B\) and nonnegative \(C\) offdiagonal entries: \(A=DB+C\).
Monotone+ matrices with only two components and their perturbations
are identified by investigating the properties of the splittings \(DB\), \(D+C\)
and \(DB+C\).
Monotone+ matrices characterized by three components are identified by means of
more involved decompositions of \(A\) or suitable
transformations of \(A\),
preserving the inversepositivity,
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
inversepositivity. The results are illustrated by numerical examples.
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