On rank range of interval matrices

TL;DR

This paper investigates the conditions under which an interval matrix contains a rank-1 matrix, determines the maximum rank within an interval matrix, and describes methods to find the rank range for matrices with up to three columns.

Contribution

It introduces a criterion for the presence of rank-1 matrices in interval matrices and provides a way to determine the maximum and possible ranks within these matrices.

Findings

Established a criterion for rank-1 inclusion in interval matrices

Determined the maximum rank of matrices within an interval matrix

Provided a method to find the rank range for matrices with up to 3 columns

Abstract

An interval matrix is a matrix whose entries are intervals in the set of real numbers. Let $p , q $ be nonzero natural numbers and let $\mu =( [m_{i,j}, M_{i,j}])_{i,j}$ be a $p \times q$ interval matrix; given a $p \times q$ matrix $A$ with entries in the set of real numbers, we say that $ A \in \mu $ if $a_{i,j} \in [m_{i,j}, M_{i,j}] $ for any $i,j$. We establish a criterion to say if an interval matrix contains a matrix of rank $1$. Moreover we determine the maximum rank of the matrices contained in a given interval matrix. Finally, for any interval matrix $\mu$ with no more than $3$ columns, we describe a way to find the range of the ranks of the matrices contained in $\mu$.