Interval matrices: realization of ranks by rational matrices

Abstract

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