Diagonal dominance and invertibility of matrices

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, 2023

Square matrices, A, strictly diagonally dominant belong to an important class of invertible matrices that have an LU decomposition. We will present in this work new non singularity criteria based on Crout's method for non strictly diagonally dominant pentadiagonal (or tridiagonal) matrices that admit an LU decomposition. These criteria are simple and easy to implement. There are many papers on this subject in the literature. However, the results that ensure non singularity of A usually depend on the conditions that are not promptly obtained.

Http Dx Doi Org 10 1080 03081087 2010 547495, 2011

For a square (0, 1, −1) sign pattern matrix S, denote the qualitative class of S by Q(S). In this paper, we investigate the relationship between sign patterns and matrices that diagonalise an irreducible nonnegative matrix. We explicitly describe the sign patterns S such that every matrix in Q(S) diagonalises some irreducible nonnegative matrix. Further, we characterise the sign patterns S such that some member of Q(S) diagonalises an irreducible nonnegative matrix. Finally, we provide necessary and sufficient conditions for a multiset of real numbers to be realised as the spectrum of an irreducible nonnegative matrix M that is diagonalised by a matrix in the qualitative class of some S 2 N S sign pattern.

SIAM Journal on Matrix Analysis and Applications, 2007

An m by n sign pattern S is an m by n matrix with entries in {+, −, 0}. Such a sign pattern allows a positive (resp., nonnegative) left inverse, provided that there exist an m by n matrix A with the sign pattern S and an n by m matrix B with only positive (resp., nonnegative) entries satisfying BA = In, where In is the n by n identity matrix. For m > n ≥ 2, a characterization of m by n sign patterns with no rows of zeros that allow a positive left inverse is given. This leads to a characterization of all m by n sign patterns with m ≥ n ≥ 2 that allow a positive left inverse, giving a generalization of the known result for the square case, which involves a related bipartite digraph. For m ≥ n, m by n sign patterns with all entries in {+, 0} and m by 2 sign patterns with m ≥ 2 that allow a nonnegative left inverse are characterized, and some necessary or sufficient conditions for a general m by n sign pattern to allow a nonnegative left inverse are presented. Key words. bipartite digraph, nonnegative left inverse, positive left inverse, positive left nullvector, sign pattern, strong Hall 1. Introduction. An m by n sign pattern S = [s ij ] is an m by n matrix with entries in {+, −, 0}. If a sign pattern S has all entries in {+, 0}, then S is a nonnegative sign pattern. A subpattern of S is an m by n sign pattern U = [u ij ] such that u ij = 0 whenever s ij = 0. If U is a subpattern of S, then S is a superpattern of U. The sign pattern class Q(S) of an m by n sign pattern S is the set of m by n matrices A = [a ij ] such that sgn(a ij ) = s ij for all i, j. If A ∈ Q(S), then A is a realization of S.

Positive Systems, 2009

A nonsingular real matrix A is said to be inverse-positive if all the elements of its inverse are nonnegative. This class of matrices contains the M-matrices, from which inherit some of their properties and applications, especially in Economy. In this work we analyze the inverse-positive concept for a particular type of pattern: the checkerboard pattern. In addition, we study the Hadamard product of certain classes of inverse-positive matrices whose entries have a particular sign pattern.

Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, 2011

In this paper, we modify Eschenbach’s algorithm for constructing sign idempotent sign pattern matrices so that it correctly constructs all of them. We find distinct classes of sign idempotent sign pattern matrices that are signature similar to an entrywise nonnegative sign pattern matrix. Additionally, if for a sign idempotent sign pattern matrix A there exists a signature matrix S such that SAS is nonnegative, we prove such S is unique up to multiplication by -1 if the signed digraph D(A) is not disconnected.

Linear Algebra and its Applications, 1998

A new nonsingularity criterion for an n × n real matrix A based on sign distribution and some conditions on the elements is given. This work was motivated by the solution of Poisson's equation for doubly connected regions. We consider a matrix A arising in the above application to demonstrate the theory.

Linear Multilinear Algebra, 1996

We investigate matrices which have a positive eigenvalue by virtue of their sign{pattern and regardless of the magnitudes of the entries. When all the o {diagonal entries are nonzero, we show that an n n sign{pattern, n 6 = 3; 4, requires a positive eigenvalue if and only if it has at least one nonnegative diagonal entry and every cycle of length greater than one in its signed digraph is positive. When n = 3; 4, or when not all o {diagonal entries are nonzero, positivity of the cycles of length greater than one is no longer necessary. In the course of proving these results we observe certain necessary and certain su cient

Electronic Journal of Linear Algebra, 2008

An m by n sign pattern A is an m by n matrix with entries in {+, −, 0}. The sign pattern A requires a positive (resp. nonnegative) left inverse provided each real matrix with sign pattern A has a left inverse with all entries positive (resp. nonnegative). In this paper, necessary and sufficient conditions are given for a sign pattern to require a positive or nonnegative left inverse. It is also shown that for n ≥ 2, there are no square sign patterns of order n that require a positive (left) inverse, and that an n by n sign pattern requiring a nonnegative (left) inverse is permutationally equivalent to an upper triangular sign pattern with positive main diagonal entries and nonpositive off-diagonal entries.

Electronic Journal of Linear Algebra, 2014

In this paper it is shown that an eventually nonnegative matrix A whose index of zero is less than or equal to one, exhibits many of the same combinatorial properties as a nonnegative matrix. In particular, there is a positive integer g such that A g is nonnegative, A and A g have the same irreducible classes, and the transitive closure of the reduced graph of A is the same as the transitive closure of the reduced graph of A g . In this instance, many of the combinatorial properties of nonnegative matrices carry over to this subclass of the eventually nonnegative matrices.

2002

In this paper it is shown that an eventually nonnegative matrix A whose index of zero is less than or equal to one, exhibits many of the same combinatorial properties as a nonnegative matrix. In particular, there is a positive integer g such that Ag is nonnegative, A and Ag have the same irreducible classes, and the transitive closure of the reduced graph of A is the same as the transitive closure of the reduced graph of Ag. In this instance, many of the combinatorial properties of nonnegative matrices carry over to this subclass of the eventually nonnegative matrices. AMS subject classifications. 15A18, 15A48, 15A21