On the nonsingularity of matrices with certain sign patterns (Q1381272)

The authors propose a nonsingularity criterion for matrices \(A= (a_{ij})_{m \times m}\) of the following two types of sign distribution: \[ \text{SD1: for }i\leq j,\;a_{ij}>0; \text{ for } i>j,\;(-1)^{i+j} a_{ij}>0; \] \[ \text{SD2: for } i\leq j,\;a_{ij} >0; \text{ for } i>j,\;(-1)^{i+j+1} a_{ij} >0 \] to be sign-nonsingular. Such matrices arise in the solution of linear systems for Poisson's equation in doubly connected regions by using difference equations or finite element methods. The criterion consists of a set of conditions which is easily computable and sufficient. A numerical example is given for demonstrating the theory.