On the maximum rank of totally nonnegative matrices

[EN] Let A is an element of R-nxn be a totally nonnegative matrix with principal rank p, that is, every minor of A is nonnegative and p is the size of the largest invertible principal submatrix of A. We introduce the sequence of the first p-indices of A as the first initial row and column indices of a p x p invertible principal submatrix of A with rank p. Then, we study the linear dependence relations between the rows and columns indexed by the sequence of the first p-indices of A and the remaining of its rows and columns. These relations, together with the irreducibility property of some submatrices of A, allow us to present an algorithm that calculates the maximum rank of A as a function of the distribution of the first p-indices. Finally, we present a method to construct n x n totally nonnegative matrices with given rank r, principal rank p and a specific sequence of the first p-indices. (C) 2018 Elsevier Inc. All rights reserved.This research was supported by the Spanish DGI grants MTM2013-43678-P, MTM2017-85669-P and MTM2017-90682-REDT.Cantó Colomina, R.; Urbano Salvador, AM. (2018). On the maximum rank of totally nonnegative matrices. Linear Algebra and its Applications. 551:125-146. https://doi.org/10.1016/j.laa.2018.03.045S12514655

Totally nonnegative matrices

An m-by-n matrix A is called totally nonnegative (resp. totally positive) if the determinant of every square submatrix (i.e., minor) of A is nonnegative (resp. positive). The class of totally nonnegative matrices has been studied considerably, and this class arises in a variety of applications such as differential equations, statistics, mathematical biology, approximation theory, integral equations and combinatorics. The main purpose of this thesis is to investigate several aspects of totally nonnegative matrices such as spectral problems, determinantal inequalities, factorizations and entry-wise products. It is well-known that the eigenvalues of a totally nonnegative matrix are nonnegative. However, there are many open problems about what other properties exist for the eigenvalues of such matrices. In this thesis we extend classical results concerning the eigenvalues of a totally nonnegative matrix and prove that the positive eigenvalues of an irreducible totally nonnegative matrix are distinct. We also demonstrate various new relationships between the sizes and the number of Jordan blocks corresponding to the zero eigenvalue of an irreducible totally nonnegative matrix. These relationships are a necessary first step to characterizing all possible Jordan canonical forms of totally nonnegative matrices. Another notion investigated is determinantal inequalities among principal minors of totally nonnegative matrices. A characterization of all inequalities that hold among products of principal minors of totally nonnegative matrices up to at most 5 indices is proved, along with general conditions which guarantee when the product of two principal minors is less than another product of two principal minors. A third component of this thesis is a study of entry-wise products of totally nonnegative matrices. In particular, we consider such topics as: closure under this product, questions related to zero/non-zero patterns, and determinantal inequalities associated with this special product. Finally, a survey of classical results and recent developments, including: commonalities and differences among totally nonnegative matrices and other positivity classes of matrices; perturbations and factorizations of totally nonnegative matrices, are discussed

On the minimum rank of the join of graphs and decomposable graphs

AbstractFor a given undirected graph G, the minimum rank of G is defined to be the smallest possible rank over all real symmetric matrices A whose (i,j)th entry is nonzero whenever i≠j and {i,j} is an edge in G. In this work we consider joins and unions of graphs, and characterize the minimum rank of such graphs in the case of ‘balanced inertia’. Several consequences are provided for decomposable graphs, also known as cographs

On the Null Space Structure Associated with Trees and Cycles

In this work, we study the structure of the null spaces of matrices associated with graphs. Our primary tool is utilizing Schur complements based on certain collections of independent vertices. This idea is applied in the case of trees, and seems to represent a unifying theory within the context of the support of the null space. We extend this idea and apply it to describe the null vectors and corresponding nullities of certain symmetric matrices associated with cycle

A novel streamlined trauma response team training improves imaging efficiency for pediatric blunt abdominal trauma patients

Background/purpose The morbidity and mortality of children with traumatic injuries are directly related to the time to definitive management of their injuries. Imaging studies are used in the trauma evaluation to determine the injury type and severity. The goal of this project is to determine if a formal streamlined trauma response improves efficiency in pediatric blunt trauma by evaluating time to acquisition of imaging studies and definitive management. Methods This study is a chart review of patients < 18 years who presented to a pediatric trauma center following blunt trauma requiring trauma team activation. 413 records were reviewed to determine if training changed the efficiency of CT acquisition and 652 were evaluated for FAST efficiency. The metrics used for comparison were time from ED arrival to CT image, FAST, and disposition. Results Time from arrival to CT acquisition decreased from 37 (SD 23) to 28 (SD27) min (p < 0.05) after implementation. The proportion of FAST scans increased from 315 (63.5%) to 337 (80.8%) and the time to FAST decreased from 18 (SD15) to 8 (SD10) min (p < 0.05). The time to operating room (OR) decreased after implementation. Conclusion The implementation of a streamlined trauma team approach is associated with both decreased time to CT, FAST, OR, and an increased proportion of FAST scans in the pediatric trauma evaluation. This could result in the rapid identification of injuries, faster disposition from the ED, and potentially improve outcomes in bluntly injured children

On Perron complements of totally nonnegative matrices

AbstractAn n×n matrix is called totally nonnegative if every minor of A is nonnegative. The problem of interest is to describe the Perron complement of a principal submatrix of an irreducible totally nonnegative matrix. We show that the Perron complement of a totally nonnegative matrix is totally nonnegative only if the complementary index set is based on consecutive indices. We also demonstrate a quotient formula for Perron complements analogous to the so-called quotient formula for Schur complements, and verify an ordering between the Perron complement and Schur complement of totally nonnegative matrices, when the Perron complement is totally nonnegative