Subtotally positive and Monge matrices

A real matrix is called k-subtotally positive if the determinants of all its submatrices of order at most k are positive. We show that for an m × n matrix, only mn inequalities determine such class for every k, 1 ? k ? min(m,n). Spectral properties of square k-subtotally positive matrices are studied. Finally, completion problems for 2-subtotally positive matrices and their additive counterpart, the anti-Monge matrices, are investigated. Since totally positive matrices are 2-subtotally positive as well, the presented necessary conditions for this completion problem are also necessary conditions for totally positive matrices.     

On visualization scaling, subeigenvectors and Kleene stars in max algebra

The purpose of this paper is to investigate the interplay arising between max algebra, convexity and scaling problems. The latter, which have been studied in nonnegative matrix theory, are strongly related to max algebra. One problem is that of strict visualization scaling, defined as, for a given nonnegative matrix A, a diagonal matrix X such that all elements of X^-1AX are less than or equal to the maximum cycle geometric mean of A, with strict inequality for the entries which do not lie on critical cycles. In this paper such scalings are described by means of the max algebraic subeigenvectors and Kleene stars of nonnegative matrices as well as by some concepts of convex geometry.     

A max version of the generalized spectral radius theorem

Let Ψ be a bounded set of n × n nonnegative matrices in max algebra. In this paper we propose the notions of the max algebra version of the generalized spectral radius μ(Ψ) of Ψ, and the max algebra version of the joint spectral radius η(Ψ) of Ψ. The max algebra version of the generalized spectral radius theorem μ(Ψ) = η(Ψ) is established. We propose the relationship between the generalized spectral radius ρ(Ψ) of Ψ (in the sense of Daubechies and Lagarias) and its max algebra version μ(Ψ). Moreover, a generalization of Elsner and van den Driessche’s lemma is presented as well.     

Interlacing properties of the eigenvalues of some matrix classes

We establish the eigenvalue interlacing property (i.e. the smallest real eigenvalue of a matrix is less than the smallest real eigenvalue of any of its principal submatrices) for the class of matrices introduced by Kotelyansky (all principal and almost principal minors of these matrices are positive). We show that certain generalizations of Kotelyansky and totally positive matrices possess this property. We also prove some interlacing inequalities for the other eigenvalues of Kotelyansky matrices.     

Intervals of totally nonnegative matrices

Totally 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).     

Almost strictly totally positive matrices

A determinantal identity, frequently used in the study of totally positive matrices, is extended, and then used to re-prove the well-known univariate knot insertion formula for B-splines. Also we introduce a class of matrices, intermediate between totally positive and strictly totally positive matrices. The determinantal identity is used to show any minor of such matrices is positive if and only if its diagonal entries are positive. Among others, this class of matrices includes B-splines collocation matrices and Hurwitz matrices.This author acknowledges a sabbatical stay at IBM T.J. Watson Research Center in 1990, which was supported by a DGICYT grant from Spain.     

On the max version of the generalized spectral radius theorem

Let Ψ be a bounded set of n×n non-negative matrices. Recently, the max algebra version μ(Ψ) of the generalized spectral radius of Ψ was introduced. We show that

     

Characterizations of rectangular totally and strictly totally positive matrices

An n×m real matrix A is said to be totally positive (strictly totally positive) if every minor is nonnegative (positive). In this paper, we study characterizations of these classes of matrices by minors, by their full rank factorization and by their thin QR factorization.     

Accurate computations with Said-Ball-Vandermonde matrices

A generalization of the Vandermonde matrices which arise when the power basis is replaced by the Said-Ball basis is considered. When the nodes are inside the interval (0,1), then those matrices are strictly totally positive. An algorithm for computing the bidiagonal decomposition of those Said-Ball-Vandermonde matrices is presented, which allows us to use known algorithms for totally positive matrices represented by their bidiagonal decomposition. The algorithm is shown to be fast and to guarantee high relative accuracy. Some numerical experiments which illustrate the good behaviour of the algorithm are included.     

The α-scalar diagonal stability of block matrices

We generalize two results: Kraaijevanger’s 1991 characterization of diagonal stability via Hadamard products and the block matrix version of the closure of the positive definite matrices under Hadamard multiplication. We restate our generalizations in terms of P^α-matrices and α-scalar diagonally stable matrices.     

Column Ranks and Their Preservers of Matrices Over Max Algebra

The column rank of an m by n matrix A over max algebra is the weak dimension of the column space of A. We compare the column rank with rank of matrices over max algebra. We also characterize the linear operators which preserve the column rank of matrices over max algebra.     

Intervals of Totally Nonnegative and Related Matrices

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.     

Equalities and inequalities for inertias of hermitian matrices with applications

The inertia of a Hermitian matrix is defined to be a triplet composed of the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we show some basic formulas for inertias of 2×2 block Hermitian matrices. From these formulas, we derive various equalities and inequalities for inertias of sums, parallel sums, products of Hermitian matrices, submatrices in block Hermitian matrices, differences of outer inverses of Hermitian matrices. As applications, we derive the extremal inertias of the linear matrix expression A-BXB^∗ with respect to a variable Hermitian matrix X. In addition, we give some results on the extremal inertias of Hermitian solutions to the matrix equation AX=B, as well as the extremal inertias of a partial block Hermitian matrix.     

Bidiagonalization of oscillatory matrices

We prove that an oscillatory matrix is similar to a bidiagonal nonnegative matrix by means of a totally positive matrix of change of basis. New characterizations of oscillatory and nonsingular totally positive matrices in terms of similarity are provided.     

Invariance properties of a triple matrix product involving generalized inverses

Given complex-valued matrices A, B and C of appropriate dimensions, this paper investigates certain invariance properties of the product AXC with respect to the choice of X, where X is a generalized inverse of B. Different types of generalized inverses are taken into account. The purpose of the paper is three-fold: First, to review known results scattered in the literature, second, to demonstrate the connection between invariance properties and the concept of extremal ranks of matrices, and third, to add new results related to the topic.     

Column Ranks and Their Preservers of Matrices Over Max Algebra

The column rank of an m by n matrix A over max algebra is the weak dimension of the column space of A. We compare the column rank with rank of matrices over max algebra. We also characterize the linear operators which preserve the column rank of matrices over max algebra.     

Notes on Hilbert and Cauchy matrices

Inspired by examples of small Hilbert matrices, the author proves a property of symmetric totally positive Cauchy matrices, called AT-property, and consequences for the Hilbert matrix.     

Invertible commutativity preservers of matrices over max algebra

The max algebra consists of the nonnegative real numbers equipped with two binary operations, maximization and multiplication. We characterize the invertible linear operators that preserve the set of commuting pairs of matrices over a subalgebra of max algebra.     

Criteria for the stability of the finiteness property and for the uniqueness of Barabanov norms

A set of matrices is said to have the finiteness property if the maximal rate of exponential growth of long products of matrices drawn from that set is realised by a periodic product. The extent to which the finiteness property is prevalent among finite sets of matrices is the subject of ongoing research. In this article, we give a condition on a finite irreducible set of matrices which guarantees that the finiteness property holds not only for that set, but also for all sufficiently nearby sets of equal cardinality. We also prove a theorem giving conditions under which the Barabanov norm associated to a finite irreducible set of matrices is unique up to multiplication by a scalar, and show that in certain cases these conditions are also persistent under small perturbations.