A generalization of Rohn's theorem on full-rank interval matrices

TL;DR

This paper extends Rohn's theorem on the nonsingularity of all matrices within a specific class of interval matrices from square matrices to more general closed interval matrices, broadening the theoretical understanding of matrix invertibility.

Contribution

It generalizes Rohn's theorem from square interval matrices to the broader class of general closed interval matrices, providing new insights into matrix invertibility.

Findings

Generalized Rohn's theorem to closed interval matrices

Characterized conditions for full-rank in broader matrix classes

Enhanced understanding of matrix invertibility within interval matrices

Abstract

A general closed interval matrix is a matrix whose entries are closed connected nonempty subsets of the set of the real numbers, while an interval matrix is defined to be a matrix whose entries are closed bounded nonempty intervals in the set of real numbers. We say that a matrix $A$ with constant entries is contained in a general closed interval matrix $\mu$ if and only if, for every $i,j$, we have that $A_{i,j} \in \mu_{i,j}$. Rhon characterized full-rank square interval matrices, that is, square interval matrices $\mu$ such that every constant matrix contained in $\mu$ is nonsingular. In this paper we generalize this result to general closed interval matrices.