# Generalization of real interval matrices to other fields

## TL;DR

This paper extends the concept of interval matrices from real numbers to other fields, especially rational numbers, and provides criteria for rank properties of these generalized matrices.

## Contribution

It introduces rational interval matrices and establishes criteria for their rank properties, generalizing existing real interval matrix results to other fields.

## Key findings

- A real interval matrix with rational endpoints contains a rational rank-one matrix if and only if it contains such a matrix.
- Criteria are provided for when a rational interval matrix contains a full-rank matrix.
- A method is described to determine the maximal rank of a matrix within a set over a field K.

## Abstract

An interval matrix is a matrix whose entries are intervals in the set of real numbers. We generalize this concept, which has been broadly studied, to other fields. Precisely we define a rational interval matrix to be a matrix whose entries are intervals in the set of the rational numbers. We prove that a (real) interval matrix with endpoints of all its entries in the set of the rational numbers contains a rank-one matrix if and only if contains a rational rank-one matrix and contains a matrix with rank smaller than min{p,q} if and only if it contains a rational matrix with rank smaller than min{p,q}; from these results and from the analogous criterions for (real) inerval matrices, we deduce immediately a criterion to see when a rational interval matrix contains a rank-one matrix and a criterion to see when it is full-rank, that is, all the matrices it contains are full-rank. Moreover, given a field K and a matrix a whose entries are subsets of K, we describe a criterion to find the maximal rank of a matrix contained in a.