Intervals of Totally Nonnegative and Related Matrices

Abstract

We consider the class of the totally nonnegative matrices, i.e., the matrices having all their minors nonnegative, and intervals of matrices with respect to the chequerboard partial ordering, which results from the usual entrywise partial ordering if we reverse the inequality sign in all components having odd index sum. For these intervals we study the following conjecture: If the left and right endpoints of an interval are nonsingular and totally nonnegative then all matrices taken from the interval are nonsingular and totally nonnegative. We present a new class of the totally nonnegative matrices for which this conjecture holds true. Similar results for classes of related matrices are also given.A class of real matrices often appearing in the iterative solution of differential equations are the inverse nonnegative matrices; these are nonsingular matrices the inverses of which are entrywise nonnegative. A special class of practical importance are the M -matrices. These are inverse nonnegative matrices with nonpositive off-diagonal entries. Kuttler has shown in [10] that the inverse nonnegative matrices enjoy a certain interval property: If A and A are inverse nonnegative and A ≤ A in the usual entrywise partial ordering then any matrix lying between both matrices is also inverse nonnegative. In the sequel we will see that a related interval property holds true for some classes of the totally nonnegative matrices, where the partial ordering is now the chequerboard partial ordering.

Equal entries in totally positive matrices

“…Here, we advance the question of which are the minimal sets of entries that must be perturbed in a given T N matrix that it become T P . Our interest stems, in part, from the likely value of this information in solving two outstanding problems: (1) the Garloff conjecture [7] and (2) the relationship between T N completable and T P completable patterns. In the latter case, author Johnson conjectures that the T N completable patterns are properly contained in the T P completable ones.…”