Search

It is shown that for positive real numbers $ 0, Comment: 9 pages

We show that every tropical totally positive matrix can be uniquely represented as the transfer matrix of a canonical totally connected weighted planar network. We deduce a uniqueness theorem for the factorization of a tropical totally positive in terms of elementary Jacobi matrices., Comment: The first author acknowledges the support of the Gaspard Monge (PGMO) program of Fondation Mathematique Hadamard, EDF, Orange and Thales, of the ICODE institute of Paris-Saclay, also of Labex Hadamard. This work started when the second author was with INRIA, being supported by the Chateaubriand program from the French Ministery of Foreign Affairs and by an INRIA fellowship

[EN] The combined matrix of a nonsingular matrix A is the Hadamard (entrywise) product . This paper deals with the characterization of the diagonal entries of a combined matrix C(A) of a given nonsingular real matrix A. A partial answer describing the diagonal entries of C(A) in the positive definite case was given by Fiedler in 1964. Recently in 2011, Fiedler and Markham characterized the sequence of diagonal entries of the combined matrix C(A) for any totally positive matrix A when the size is 3. For this case, we characterize totally negative matrices and we find necessary and sufficient conditions for the sequence of diagonal entries of C(A), in both cases, symmetric and nonsymmetric., This work was supported by Spanish DGI [grant number MTM-2014-58159-P]; Dominican Republic FONDOCYT [grant number 2015-1D2-166].

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.

We present a new class of totally positive Toeplitz matrices composed of recently introduced hyperfibonacci numbers of the ?$r$?-th generation. As a consequence, we obtain that all sequences ?$F_n^{(r)}$? of hyperfibonacci numbers of ?$r$?-th generation are log-concave for ?$r \ge 1$? and large enough ?$n$?. Predstavimo nov razred totalno pozitivnih Toeplitzevih matrik, sestavljenih iz nedavno vpeljanih hiperfibonaccijevih števil ?$r$?-te generacije. Pokažemo, da so vsa zaporedja ?$F_n^{(r)}$? hiperfibonaccijevih števil ?$r$?-te generacije log-konkavna za ?$r \ge 1$? in dovolj velike ?$n$?.

A frame for an N-dimensional Hilbert space H is said to be a full spark frame if any N of its vectors form a basis for H. In this talk we present a characterization of full spark frames in terms of certain totally nonsingular matrices. Also, we construct a family of infinite totally positive matrices which are a subclass of the class of totally nonsingular matrices. This is a joint work with Damir Bakić.

The intersection between the set of totally nonnegative matrices, which are of interest in many areas of matrix theory and its applications, and the set of density matrices, which provide the mathematical description of quantum states, are investigated. The single qubit case is characterized, and several equivalent conditions for a quantum channel to preserve the set in that case are given. Higher dimensional cases are also discussed.

In this paper, we construct one of the forms of totally positive Toeplitz matrices from upper or lower bidiagonal totally nonnegative matrix. In addition, some properties related to this matrix involving its factorization are presented.

A matrix is called totally positive if every minor of it is positive. Such matrices are well studied and have numerous applications in Mathematics and Computer Science. We study how many times the value of a minor can repeat in a totally positive matrix and show interesting connections with incidence problems in combinatorial geometry. We prove that the maximum possible number of repeated $d \times d$-minors in a $d \times n$ totally-positive matrix is $O(n^{d-\frac{d}{d+1}})$. For the case $d=2$ we also show that our bound is optimal. We consider some special families of totally postive matrices to show non-trivial lower bounds on the number of repeated minors. In doing so, we arrive at a new interesting problem: How many unit-area and axis-parallel rectangles can be spanned by two points in a set of $n$ points in the plane? This problem seems to be interesting in its own right especially since it seem to have a flavor of additive combinatorics and relate to interesting incidence problems where considering only the topology of the curves involved is not enough. We prove an upper bound of $O(n^{\frac{4}{3}})$ and provide a lower bound of $n^{1+\frac{1}{O(\log\log n)}}$., Comment: 11 pages

We consider the problem of approximating matrices by matrices of rank n in the | · | p , q norm. Among other results, we prove that if A is a totally positive matrix, and | A | 1 , 1 is the norm given by the sum of the absolute values of the entries of the matrix, then a best rank n approximation to A in this norm is given by a matrix that agrees with A on n rows and n columns.

We survey the history of totally positive matrices and the generalization to Lie groups. We describe a reduction of a bilinear form to a canonical form (generalizing the case of symplectic nondegenerate forms) using ideas from total positivity; we also place this in a Lie group context. We give a short exposition of results of Fock and Goncharov on the study of homomorphisms of the fundamental group of a closed surface into a Lie group., Comment: 8 pages

Total positivity (TP) is a powerful concept that arises in various branches of mathematics. A matrix $M = [m_{; ; i, j}; ; ]_{; ; i, j \ge 0}; ; $ is {; ; \it totally positive}; ; if all its minors are non-negative real numbers. A remarkable spectral property of an $n \times n$ totally positive matrix $M$ is that $M$ has $n$ distinct positive eigenvalues. Any totally positive matrix can be relalized as a matrix of generalized complete homogeneous symmetric functions evaluated at non-negative real numbers. Total positivity of some types of triangular matrices, including {; ; \it Catalan triangles}; ; by Aigner and by Shapiro can be proved using the Aissen-Schoenberg-Whitney Theorem. On the base of these ideas, we demonstrate TP of some combinatorial matrices, indluding Stirling numbers of the first and the second kind. In addition, we present a class of totally positive matrices, defined by means of hypersequences that are recently introduced by Dil and Mez\H o.

Recently Chen-Liang-Wang (Linear Algerbra Appl. 471 (2015) 383—393) proved some sufficient conditions for the total positivity of Catalan-Stieltjes matrices. Our aim is to provide a combinatorial interpretation of their sufficiant conditions. More precisely, for any Catalan-Stieltjes matrix $A$ we construct a digraph with a weight, which is positive under their sufficient conditions, such that every minor of $A$ is equal to the sum of weights of families of nonintersecting paths of the digraph. We have also an analogue result for the minors of Hankel matrix associated to the first column of Catalan-Stieltjes matrix $A$.

Here, we define and consider (linear) TP-directions and TP-paths for a totally nonnegative matrix, in an effort to more deeply understand perturbation of a TN matrix to a TP matrix. We give circumstances in which a TP-direction exists and an example to show that they do not always exist. A strategy to give (nonlinear) TP-paths is given (and applied to this example). A long term goal is to understand the sparsest TP-perturbation for application to completion problems., Supported by FEDER Funds through Programa Operacional Factores de Competitividade - COMPETE and by Portuguese Funds through FCT - “Fundação para a Ciência e a Tecnologia”, within the Project PEst-OE/MAT/UI0013/2014.

A fast and accurate algorithm for solving a Bernstein–Vandermonde linear system is presented. The algorithm is derived by using results related to the bidiagonal decomposition of the inverse of a totally positive matrix by means of Neville elimination. The use of explicit expressions for the determinants involved in the process serves to make the algorithm both fast and accurate.

It has long been known that the eigenvalues of a totally positive matrix interlace the eigenvalues of its maximal leading principal submatrix. Motivated by recent questions arising from studying the roots of certain biorthogonal polynomials, we extend the classical strict interlacing fact to other classes of totally nonnegative matrices. Mathematics subject classification (2000): 15A48, 15A13, 33C45.

We consider the problem of performing accurate computations with rectangular $(m\times n)$ totally nonnegative matrices. The matrices under consideration have the property of having a unique representation as products of nonnegative bidiagonal matrices. Given that representation, one can compute the inverse, LDU decomposition, eigenvalues, and SVD of a totally nonnegative matrix to high relative accuracy in $\mathcal{O}(\max(m^3,n^3))$ time—much more accurately than conventional algorithms that ignore that structure. The contribution of this paper is to show that the high relative accuracy is preserved by operations that preserve the total nonnegativity—taking a product, re-signed inverse (when $m=n$), converse, Schur complement, or submatrix of a totally nonnegative matrix, any of which costs at most $\mathcal{O}(\max(m^3,n^3))$. In other words, the class of totally nonnegative matrices for which we can do numerical linear algebra very accurately in $\mathcal{O}(\max(m^3,n^3))$ time (namely, those for which we have a product representation via nonnegative bidiagonals) is closed under the operations listed above.

In this paper, we construct a matrix, which transforms a generalized C-Bezier basis into a generalized uniform algebraic-trigonometric B-spline (C-B-spline or UAT B-spline) basis. We also show that it is a totally positive matrix and give a normalized B-basis of the generalized UAT B-splines.

A matrix is called totally positive if all its minors are positive. If a totally positive matrix A is partitioned as $A = (A_{ij})$ $i,j=1,2,\ldots,k$, in which each block $A_{ij}$ is n x n, we show that the k x k compressed matrix given by $(\det A_{ij})$ is also totally positive and that the determinant of the compressed matrix exceeds A when k=2,3. An extension that allows for overlapping blocks is also presented when k=2,3. For $k \geq 4$ we verify, by example, that the k x k compressed matrix of a totally positive matrix need not be totally positive.

Let G be a Lie group, let L(G) be its Lie algebra, and let exp : denote the exponential mapping. For , we define the tangent set of S by . We say that a semigroup S is strictly infinitesimally generated if S is the same as the semigroup generated by exp(L(S)). We find a tangent set of the semigroup of all non-singular totally positive matrices and show that the semigroup is strictly infinitesimally generated by the tangent set of the semigroup. This generalizes the familiar relationships between connected Lie subgroups of G and their Lie algebras

We consider the class of totally nonnegative (TN) matrices---matrices all of whose minors are nonnegative. Any nonsingular TN matrix factors as a product of nonnegative bidiagonal matrices. The entries of the bidiagonal factors parameterize the set of nonsingular TN matrices. We present new O(n3) algorithms that, given the bidiagonal factors of a nonsingular TN matrix, compute its eigenvalues and SVD to high relative accuracy in floating point arithmetic, independent of the conventional condition number. All eigenvalues are guaranteed to be computed to high relative accuracy despite arbitrary nonnormality in the TN matrix. We prove that the entries of the bidiagonal factors of a TN matrix determine its eigenvalues and SVD to high relative accuracy. We establish necessary and sufficient conditions for computing the entries of the bidiagonal factors of a TN matrix to high relative accuracy, given the matrix entries. In particular, our algorithms compute all eigenvalues and the SVD of TN Cauchy, Vandermonde, Cauchy--Vandermonde, and generalized Vandermonde matrices to high relative accuracy.

It is shown that for two real vectors x , y there is a totally positive matrix A such that Ax = y if and only if a certain refinement of a classical (sign) variation diminishing relationship between x and y is satisfied.

Between any two rows (columns) of an m-by-n totally nonnegative matrix polynomial, it is not difficult to show that a new polynomial row (column) may be inserted to form a larger, totally nonnegative matrix polynomial. The analogous question, in which 'totally nonnegative' is replaced by 'totally positive' arises in completion problems and the extension of collocation matrices, and its answer is far less clear. Here the totally positive question is answered affirmatively, and by similar techniques affirmative answers may also be given for several other classes of matrix functions.

An n × m real matrix A is a totally positive matrix if all its minors are nonnegative. The Neville elimination process is studied in relation to the existence of a totally positive factorization LS of a rectangular matrix. An LS factorization is obtained for a totally positive matrix, where L is a lower echelon form matrix, of size n × k, and S is an upper echelon form matrix, of size k × m, and both L and S are totally positive matrices.

In this paper we prove that Neville elimination can be matricially described by elementary matrices. A PLU-factorization is obtained for any n×m matrix, where P is a permutation matrix, L is a lower triangular matrix (product of bidiagonal factors) and U is an upper triangular matrix.This result generalizes the Neville factorization usually applied to characterize the totally positive matrices. We prove that this elimination procedure is an alternative to Gaussian elimination and sometimes provides a lower computational cost.

A matrix is totally positive (respectively, strictly totally positive) if all its minors are nonnegative (respectively, positive). In this paper, we revisit the main determinantal criteria for these matrices and provide two new ones: one of them for the total positivity of m×n matrices with any n consecutive rows linearly independent and another one to check if an n×n totally positive matrix has all minors of order less than k (k⩽n) positive.

We establish a sufficient condition for strict total positivity of a matrix In particular, we show that if the (positive) elements of a square matrix grow sufficiently fast as their distance from the diagonal of the matrix increases, then the matrix is strictly totally positive.

A real matrix is said to be totally positive if all its minors are nonnegative. In this paper it is shown that properties of totally positive matrices can be applied to graph theory, and conversely. In fact, some properties of undirected and directed graphs (digraphs) are characterized in terms of the associated totally positive matrices. Some results on the existence of nonintersecting paths in a digraph are also provided.

Let A = (a(ij)) is an element of R-nxm be a totally nonpositive matrix with rank(A) = r, This research was supported by the Spanish DGI grant MTM2010-18228.

Totally nonnegative matrices, i.e., matrices having all minors nonnegative, are con- sidered. A condensed form of the Cauchon algorithm which has been proposed for finding a param- eterization 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 determi- nantal 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.

A study of the maximum number of equal entries in totally positive and totally nonsingular m-by-n, matrices for small values of m and n, is presented. Equal entries correspond to entries of the totally nonnegative matrix J that are not changed in producing a TP or TNS matrix. It is shown that the maximum number of equal entries in a 7-by-7 totally positive matrix is strictly smaller than that for a 7-by-7 totally non-singular matrix, but, this is the first pair (m; n) for which these maximum numbers differ. Using point-line geometry in the projective plane, a family of values for (m; n) for which these maximum numbers differ is presented. Generalization to the Hadamard core, as well as larger projective planes is also established. Finally, the relationship with C4 free graphs, along with a method for producing symmetric TP matrices with maximal symmetric arrangements of equal entries is discussed.