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.
Abstract. Totally nonnegative matrices, i.e., matrices having all minors nonnegative, are considered. A condensed form of the Cauchon algorithm which has been proposed for finding a parameterization of the set of these matrices with a fixed pattern of vanishing minors is derived. The close connection of this variant to Neville elimination and bidiagonalization is shown and new determinantal tests for total nonnegativity are developed which require much fewer minors to be checked than for the tests known so far. New characterizations of some subclasses of the totally nonnegative matrices as well as shorter proofs for some classes of matrices for being (nonsingular and) totally nonnegative are derived.
We consider classes of n-by-n sign regular matrices, i.e. of matrices with the property that all their minors of fixed order k have one specified sign or are allowed also to vanish, k = 1, . . . , n. If the sign is nonpositive for all k, such a matrix is called totally nonpositive. The application of the Cauchon algorithm to nonsingular totally nonpositive matrices is investigated and a new determinantal test for these matrices is derived. Also matrix intervals with respect to the checkerboard ordering are considered. This order is obtained from the usual entry-wise ordering on the set of the n-by-n matrices by reversing the inequality sign for each entry in a checkerboard fashion. For some classes of sign regular matrices, it is shown that if the two bound matrices of such a matrix interval are both in the same class then all matrices lying between these two bound matrices are in the same class, too.
Associated to a graph G is a set S(G) of all real-valued symmetric matrices whose off-diagonal entries are non-zero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be chosen. If G has n vertices, then the multiplicities of the eigenvalues of any matrix in S(G) partition n; this is called a multiplicity partition.We study graphs for which a multiplicity partition with only two integers is possible. The graphs G for which there is a matrix in S(G) with partitions [n − 2, 2] have been characterized. We find families of graphs G for which there is a matrix in S(G) with multiplicity partition [n − k, k] for k ≥ 2. We focus on generalizations of the complete multipartite graphs. We provide some methods to construct families of graphs with given multiplicity partitions starting from smaller such graphs. We also give constructions for graphs with matrix in S(G) with multiplicity partition [n − k, k] to show the complexities of characterizing these graphs.
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