Generalization of real interval matrices to other fields

Abstract

An interval matrix is a matrix whose entries are intervals in R. 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 Q. We prove that a (real) interval p × q matrix with the endpoints of all its entries in Q 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 α α α whose entries are subsets of K, we describe a criterion to find the maximal rank of a matrix contained in α α α.2010 Mathematical Subject Classification: 15A99, 15A03, 65G40

Interval matrices: realization of ranks by rational matrices

“…Moreover, in [21] we observed (see in Remark 13 there) that from the papers [3], [23] and [9] we can deduce that it is not true that, for any r, if an interval matrix with the endpoints of all its entries in Q contains a rank-r real matrix, then it contains a rank-r rational matrix.…”