Certification of the Sign Regularity of Matrix Intervals

Abstract A sign regular matrix is a matrix having the property that its non-zero minors of all orders have, for each order, an identical sign. The most important instances are the totally nonnegative matrices; these are matrices having only nonnegative minors. In this paper, intervals of real matrices with respect to the usual entry-wise partial ordering are considered. The objective is to determine the minimum number of sign regular vertex matrices with an identical signature of their minors from which it can be inferred that all element matrices in the interval are sign regular with the same signature of their minors. Results obtained so far are surveyed and approaches toward resolving a long-standing open problem are discussed.