Generalization of real interval matrices to other fields

An interval matrix is a matrix whose entries are intervals in the set of the real numbers R. This concept, which has been broadly studied, is generalized to other fields. Precisely, a rational interval matrix is defined to be a matrix whose entries are intervals in the set of the rational numbers Q. It is proved that a (real) interval p x q matrix with the endpoints of all its entries in Q contains a rank-one matrix if and only if it 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, 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, are deduced immediately. Moreover, given a field K and a matrix alpha whose entries are subsets of K, a criterion to find the maximal rank of a matrix contained in alpha is described.