A factorization theorem for banded matrices

In this paper we prove a factorization theorem for strictly m-banded totally positive matrices. We show that such a matrix is a product of m one-banded matrices with positive entries.     

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.     

The LU-factorization of totally positive matrices

An n × n real matrix A is an STP (strictly totally positive) matrix if all its minors are strictly positive. An n × n real triangular matrix A is a ΔSTP matrix if all its nontrivial minors are strictly positive. It is proved that A is an STP matrix iff A = LU where L is a lower triangular matrix, U is an upper triangular matrix, and both L and U are ΔSTP matrices. Several related results are proved. These results lead to simple proofs of many of the determinantal properties of STP matrices.     

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.     

Possible spectra of totally positive matrices

For any given set S of n distinct positive numbers, we construct a symmetric n-by-n (strictly) totally positive matrix whose spectrum is S. Thus, in order to be the spectrum of an n-by-n totally positive matrix, it is necessary and sufficient that n numbers be positive and distinct.     

Surjectivity and invertibility properties of totally positive matrices

A new “finite section” type theorem is used to show that the members of an interesting class of bounded totally positive matrices map l_∞ onto l_∞ if and only if their range contains a vector which alternates in sign and has coordinates bounded away from zero. The class of matrices studied contains all banded totally positive matrices, and thus all infinite spline collocation matrices. Connections to related work and extension to matrices which are not sign regular are indicated.     

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 the characterization of almost strictly totally positive matrices

A nonsingular matrix is called almost strictly totally positive when all its minors are nonnegative and, furthermore, these minors are strictly positive if and only if their diagonal entries are strictly positive. Almost strictly totally positive matrices are useful in Approximation Theory and Computer Aided Geometric Design to generate bases of functions with good shape preserving properties. In this paper we give an algorithmic characterization of these matrices. Moreover, we provide a determinantal characterization of them in terms of the positivity of a very reduced number of their minors and also in terms of their factorizations.     

On the characterization of almost strictly totally positive matrices

A nonsingular matrix is called almost strictly totally positive when all its minors are nonnegative and, furthermore, these minors are strictly positive if and only if their diagonal entries are strictly positive. Almost strictly totally positive matrices are useful in Approximation Theory and Computer Aided Geometric Design to generate bases of functions with good shape preserving properties. In this paper we give an algorithmic characterization of these matrices. Moreover, we provide a determinantal characterization of them in terms of the positivity of a very reduced number of their minors and also in terms of their factorizations. Both authors were partially supported by the DGICYT Spain Research Grant PB93-0310     

Indecomposable totally positive numbers in real quadratic orders

In a totally real number field, every totally positive integral number is a finite sum of (additively) indecomposable totally positive integral numbers, and up to multiplication by totally positive units, there exist only finitely many indecomposables. In the paper it is shown that in quadratic fields all these numbers can be listed in a very efficient way by using the so-called intermediate convergents of a certain quadratic irrationality. The method can be viewed as a simple extension of the standard method of calculating the fundamental unit by using continued fractions. As an application it is shown that for instance in Z|√d| a number is decomposable if its norm is >d. It is remarkable that this bound does not depend on the size of the fundamental unit.     

Subdirect sums of P(P0)-matrices and totally nonnegative matrices

In this paper, the problem of when the sub-direct sum of two strictly diagonally dominant P-matrices is a strictly diagonally dominant P-matrix is studied. In particular, it is shown that the subdirect sum of overlapping principal submatrices of strictly diagonally dominant P-matrices is a strictly diagonally dominant P-matrix. It is also established that the 2-subdirect sum of two totally nonnegative matrices is a totally nonnegative matrix under some conditions. It is obtained that a partial totally nonnegative matrix, whose graph of the specified entries is a monotonically labeled 2-chordal graph, has a totally nonnegative completion. Finally, a positive answer to the question (IV) in Fallat and Johnson [Shaun M. Fallat, C.R. Johnson, J.R. Torregrosa, A.M. Urbano, P-matrix completions under weak symmetry assumptions, Linear Algebra Appl. 312 (2000) 73-91] is given for P₀-matrices.     

On the representability of totally unimodular matrices on bidirected graphs

We present a polynomial time algorithm to construct a bidirected graph for any totally unimodular matrix B by finding node-edge incidence matrices Q and S such that QB=S. Seymour’s famous decomposition theorem for regular matroids states that any totally unimodular (TU) matrix can be constructed through a series of composition operations called k-sums starting from network matrices and their transposes and two compact representation matrices B₁,B₂ of a certain ten element matroid. Given that B₁,B₂ are binet matrices we examine the k-sums of network and binet matrices. It is shown that thek-sum of a network and a binet matrix is a binet matrix, but binet matrices are not closed under this operation for k=2,3. A new class of matrices is introduced, the so-called tour matrices, which generalise network, binet and totally unimodular matrices. For any such matrix there exists a bidirected graph such that the columns represent a collection of closed tours in the graph. It is shown that tour matrices are closed under k-sums, as well as under pivoting and other elementary operations on their rows and columns. Given the constructive proofs of the above results regarding the k-sum operation and existing recognition algorithms for network and binet matrices, an algorithm is presented which constructs a bidirected graph for any TU matrix.     

Constrained core solutions for totally positive games with ordered players

In many applications of cooperative game theory to economic allocation problems, such as river-, polluted river- and sequencing games, the game is totally positive (i.e., all dividends are nonnegative), and there is some ordering on the set of the players. A totally positive game has a nonempty core. In this paper we introduce constrained core solutions for totally positive games with ordered players which assign to every such a game a subset of the core. These solutions are based on the distribution of dividends taking into account the hierarchical ordering of the players. The Harsanyi constrained core of a totally positive game with ordered players is a subset of the core of the game and contains the Shapley value. For special orderings it coincides with the core or the Shapley value. The selectope constrained core is defined for acyclic orderings and yields a subset of the Harsanyi constrained core. We provide a characterization for both solutions.     

Structure of invertible (bi)infinite totally positive matrices

An l_∞-invertible nonfinite totally positive matrix A is shown to have one and only one “main diagonal.” This means that exactly one diagonal of A has the property that all finite sections of A principal with respect to this diagonal are invertible and their inverses converge boundedly and entrywise to A^-1. This is shown to imply restrictions on the possible shapes of such a matrix. In the proof, such a matrix is also shown to have an l_∞-invertible LDU factorization. In addition, decay of the entries of such a matrix away from the main diagonal is demonstrated. It is also shown that a bounded sign-regular matrix carrying some bounded sequence to a uniformly alternating sequence must have all its columns in c₀.     

Bidiagonal decomposition of totally positive Bernstein-Vandermonde matrices

The class of Bernstein-Vandermonde matrices (a generalization of Vandermonde matrices arising when the monomial basis is replaced by the Bernstein basis) is considered. A convenient ordering of their rows makes these matrices strictly totally positive. By using results related to total positivity and Neville elimination, an algorithm for computing the bidiagonal decomposition of a Bernstein-Vandermonde matrix is constructed. The use of explicit expressions for the determinants involved in the process serves to make the algorithm both fast and accurate. One of the applications of our algorithm is the design of fast and accurate algorithms for solving Lagrange interpolation problems when using the Bernstein basis, an approach useful for the field of Computer Aided Geometric Design since it avoids the stability problems involved with basis transformations between the Bernstein and the monomial bases. A different application consists of the use of the bidiagonal decomposition as an intermediate step of the computation of the eigenvalues and the singular value decomposition of a totally positive Bernstein-Vandermonde matrix. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Monotone positive stable matrices

The class of real matrices which are both monotone (inverse positive) and positive stable is investigated. Such matrices, called N-matrices, have the well-known class of nonsingular M-matrices as a proper subset. Relationships between the classes of N-matrices, M-matrices, nonsingular totally nonnegative matrices, and oscillatory matrices are developed. Conditions are given for some classes of matrices, including tridiagonal and some Toeplitz matrices, to be N-matrices.     

A new look at totally positive matrices

A close relationship between the class of totally positive matrices and anti-Monge matrices is used for suggesting a new direction for investigating totally positive matrices. Some questions are posed and a partial answer in the case of Vandermonde-like matrices is given.     

Accurate computations of matrices with bidiagonal decomposition using methods for totally positive matrices

A class of sign‐symmetric P‐matrices including all nonsingular totally positive matrices and their inverses as well as tridiagonal nonsingular H‐matrices is presented and analyzed. These matrices present a bidiagonal decomposition that can be used to obtain algorithms to compute with high relative accuracy their singular values, eigenvalues, inverses, or their LDU factorization. Copyright © 2012 John Wiley & Sons, Ltd.     

On totally positive sequences and functions

Let a>0 be a fixed number. A function f:R→R is said to be a-shift-generating (a-SG) if for every x∈R, is a totally positive sequence and it does not coincide with a sequence of the form , where A?0 and λ>0. In this paper, we describe all a-SG functions and obtain a new characterization of totally positive functions in the terms of a-SG functions. In addition, using characteristic properties of a-SG functions, we generalize the famous Jacobian identity in theory of elliptic functions.     

Full spark frames and totally positive matrices

In this paper we establish a connection between full spark frames and totally nonsingular matrices. Then we provide a method for constructing infinite totally positive matrices which make up a subclass of the class of totally nonsingular matrices. Using this method we then construct a family of infinite totally positive matrices parameterized by non-negative numbers which contains, as the simplest case, the infinite Pascal matrix. The paper ends with some examples and comments on full spark frames.