Necessary and sufficient conditions for regularity of interval parametric matrices

TL;DR

This paper establishes necessary and sufficient conditions for the regularity of interval parametric matrices, improving analysis especially for large parameter intervals, and applies these conditions to solve related problems.

Contribution

It introduces a novel set of conditions based on boundary hypersurfaces and spectral properties, advancing the understanding of matrix regularity in interval parametric matrices.

Findings

Conditions based on boundary hypersurfaces are effective for regularity analysis.

The methodology simplifies high-dimensional problems to lower-dimensional ones.

Applications include interval hull solutions and regularity radius calculations.

Abstract

Matrix regularity is a key to various problems in applied mathematics. The sufficient conditions, used for checking regularity of interval parametric matrices, usually fail in case of large parameter intervals. We present necessary and sufficient conditions for regularity of interval parametric matrices in terms of boundary parametric hypersurfaces, parametric solution sets, determinants, real spectral radiuses. The initial n-dimensional problem involving K interval parameters is replaced by numerous problems involving 1<= t <= min(n-1, K) interval parameters, in particular t=1 is most attractive. The advantages of the proposed methodology are discussed along with its application for finding the interval hull solution to interval parametric linear system and for determining the regularity radius of an interval parametric matrix.