Some properties of GM-matrices and their inverses
This article studies the relation between the cycle structure of the signed digraph of a matrix and the signs of its principal minors. An \(n\times n\) matrix A is a GM-matrix if its diagonal elements are negative and each negative cycle is contained in any positive cycle it intersects. (The sign of a cycle is the product of the signs of its arcs.) The matrix is consistently signed if each \(p\times p\) principal minor has sign \((- 1)^ p.\) The authors classify irreducible GM-matrices as follows: I. All cycles are negative (L-matrix). II. All cycles are positive (simply related to Z-matrices and M-matrices). III. There are positive and negative cycles, and every positive cycle has length n (degenerate case). IV. There are positive and negative cycles, and some positive cycle has length less than n. Conditions are obtained for a matrix of class III or IV to be consistently signed, and a canonical form is specified for class IV. For consistently signed class IV matrices, some information about elements of the inverse is found.
scientific article; zbMATH DE number 4077451