2 - L-matrices

Summary

Signings

Let A be an m by n matrix. Recall that A is an L-matrix if and only if every matrix in the qualitative class Q(A) has linearly independent rows. If A is an L-matrix, then every matrix obtained from A by appending column vectors is also an L-matrix. If A is an L-matrix and each of the m by n – 1 matrices obtained from A by deleting A column is not an L-matrix, then A is called A barely L-matrix [1]. Thus A barely L-matrix is an L-matrix in which every column is essential. If A is an L-matrix, then we can obtain A barely L-matrix by deleting certain columns of A. An SNS-matrix, that is, A square L-matrix, is A barely L-matrix. But there are barely L-matrices which are not square. The 3 by 4 matrix (1.10) is an L-matrix, and it follows from Theorem 1.2.5 that each of its submatrices of order 3 is not an SNS-matrix. Hence (1.10) is A barely L-matrix.

A signing of order k: is A nonzero (0, 1, – 1)-diagonal matrix of order k. A strict signing is A signing that is invertible. Let D = diag(d₁, d₂,…, d_k) be A signing of order k with diagonal entries d₁, d₂,…,d_k. If k = m, then the matrix DA is A row signing of the matrix A, and if D is A strict signing, then DA is A strict row signing of A.