Combinatorial Matrix Theory

Abstract

This book, first published in 1991, is devoted to the exposition of combinatorial matrix theory. This subject concerns itself with the use of matrix theory and linear algebra in proving results in combinatorics (and vice versa), and with the intrinsic properties of matrices viewed as arrays of numbers rather than algebraic objects in themselves. There are chapters dealing with the many connections between matrices, graphs, digraphs and bipartite graphs. The basic theory of network flows is developed in order to obtain existence theorems for matrices with prescribed combinatorial properties and to obtain various matrix decomposition theorems. Other chapters cover the permanent of a matrix, and Latin squares. The final chapter deals with algebraic characterizations of combinatorial properties and the use of combinatorial arguments in proving classical algebraic theorems, including the Cayley-Hamilton Theorem and the Jordan Canonical Form. The book is sufficiently self-contained for use as a graduate course text, but complete enough for a standard reference work on the basic theory. Thus it will be an essential purchase for combinatorialists, matrix theorists, and those numerical analysts working in numerical linear algebra.

Powers of matrices over distributive lattices?a review

“…Let us also note here, that digraphs are used in a similar way for the study of nonnegative matrices, for which a good reference is the monograph [7].…”

Mixed Matrices: Irreducibility and Decomposition

“…With respect to the admissible transformation (4) we can define the notion of irreducibility for LM-matrices, which is an extension of the well-studied concept of full indecomposability [5], [49]. First recall that a matrix A is said to be partially decomposable if it contains a zero submatrix A [I, J] = O with |I| + |J| = max(|Row(A )|, |Col(A )|); otherwise, it is called fully indecomposable.…”

“…Then A = T and hence all the nonzero entries are algebraically independent. Such a matrix is called a generic matrix in Brualdi-Ryser [5]. The admissible transformation (4) reduces toĀ = P r AP c , involving permutations only, and the LM-irreducibility is nothing but the full indecomposability.…”

“…This is called the Dulmage-Mendelsohn decomposition, or the DM-decomposition for short. See [4], [5], [8], [21], [28], [33], [37] for more about the DM-decomposition.…”

“…Ore, and others [5], [20], [21], [53]. Note also that the function γ(I, J) − |J| (with I fixed) is called the surplus function in Lovász-Plummer [21].…”