Intervals of totally nonnegative matrices

Abstract

a r t i c l e i n f o a b s t r a c t Keywords:Totally nonnegative matrix Checkerboard ordering Matrix interval Cauchon diagram Cauchon AlgorithmTotally nonnegative matrices, i.e., matrices having all their minors nonnegative, and matrix intervals with respect to the checkerboard ordering are considered. It is proven that if the two bound matrices of such a matrix interval are nonsingular and totally nonnegative (and in addition all their zero minors are identical) then all matrices from this interval are also nonsingular and totally nonnegative (with identical zero minors). IntroductionA real matrix is called totally nonnegative and totally positive if all its minors are nonnegative and positive, respectively. Such matrices arise in a variety of ways in mathematics and its applications. For background information the reader is referred to the recently published monographs [4,13]. In this paper we solve the conjecture posed by the second author in this journal in 1982 [5], see also [4, Section 3.2] and [13, Section 3.2]. This conjecture concerns the checkerboard ordering which is obtained from the usual entry-wise ordering in the set of the square real matrices of fixed order by reversing the inequality sign for each entry in a checkerboard fashion. The conjecture states that if the two bound matrices of an interval with respect to this ordering are nonsingular and totally nonnegative then all matrices lying between the two bound matrices are nonsingular and totally nonnegative, too.

Intervals of special sign regular matrices

“…In [25], we provide some further signatures for N sS R matrices which allow the interval property. These results together with the results in [4,22,23] on the interval property of some other classes of N sS R matrices evoke the (open) question whether the interval property holds for general N sS R matrices. We mention the following partial answer to this question [26] …”

“…For background information, the reader is referred to the monographs. [2,3] In [4], we apply the Cauchon algorithm [5,6] to totally nonnegative matrices and prove a long-standing conjecture posed by the second author on intervals of nonsingular totally nonnegative matrices. The underlying ordering is the checkerboard ordering which is obtained from the usual entry-wise ordering in the set of the square real matrices of fixed order by reversing the inequality sign for each entry in a checkerboard fashion.…”

Invariance of total positivity of a matrix under entry-wise perturbation and completion problems

“…If in an index sequence two indices coincide then the respective matrix has to be removed from the listing. In the following cases only the numerator matrices are listed 1 . The cases (−) and (+) refer to the −-and +-sign in (4.1).…”

“…., a r+1,1 . For a fixed entry we consider the submatrices which have the chosen entry as the only unspecified entry, viz in position (1,1). For each such submatrix we can find a positive number such that the matrix is T P .…”

“…The second author conjectured in 1982 [8] that this is true for the nonsingular totally nonnegative matrices and the so-called checkerboard ordering, see also [5,Section 3.2] and [11,Section 3.2]. In [1] we apply the so-called Cauchon algorithm, see, e.g., [3], to settle this conjecture.…”