A new characterization of inverse-positive matrices (Q810610)

An \(n\times n\) matrix M is said to be inverse-positive if \(M^{-1}\) exists and \(M^{-1}\geq 0.\) A positive splitting of a matrix M is an expression of it in the form \(M=B-A,\) \(B\geq 0,\) \(A\geq 0.\) A positive splitting \(M=B-A\) is said to be a B-splitting if B is non-singular and (a) \(Bx\geq 0\Rightarrow Ax\geq 0;\) (b) for all \(x\in {\mathbb{R}}^ n\), \(x^ T(M^ T,B^ T)\geq 0^ T\Rightarrow x\geq 0.\) The main result is that M is inverse-positive if and only if M allows a B-splitting \(M=B-A\) for which \(\mu^*<1,\) where \(\mu^*\) is the spectral radius of \(AB^{-1}\). This result generalizes the well-known one for a matrix M of the form \(M=sI-A\) where \(s>0,\) \(A\geq 0.\)