Interval matrices: realization of ranks by rational matrices

TL;DR

This paper proves that for certain interval matrices with rational endpoints, the existence of a real matrix of a specific rank guarantees the existence of a rational matrix of the same rank within the interval.

Contribution

It establishes that for particular ranks, the presence of a real matrix in an interval matrix implies the presence of a rational matrix of the same rank.

Findings

If an interval matrix contains a rank-r real matrix for r in {2, q-2, q-1, q}, then it also contains a rank-r rational matrix.

The result applies to p x q interval matrices with rational endpoints and p ≥ q.

The proof bridges the existence of real and rational matrices of specific ranks within interval matrices.

Abstract

Let $\alpha $ be a $p \times q$ interval matrix with $p \geq q$ and with the endpoints of all its entries in the set of the rational numbers. We prove that, if $\alpha $ contains a rank-$r$ real matrix with $r \in \{2, q-2,q-1,q\}$, then it contains a rank-$r$ rational matrix.