M-matrices with prescribed elementary divisors. (23rd August 2017) - Soto, Ricardo L; Díaz, Roberto C; Salas, Mario; Rojo, Oscar
Abstract: A real matrix A is said to be an M -matrix if it is of the form A = α I − B, where B is a nonnegative matrix with Perron eigenvalue ρ ( B ), and α ⩾ ρ ( B ) . This paper provides sufficient conditions for the existence and construction of an M -matrix A with prescribed elementary divisors, which are the characteristic polynomials of the Jordan blocks of the Jordan canonical form of A . This inverse problem on M -matrices has not been treated until now. We solve the inverse elementary divisors problem for diagonalizable M -matrices and the symmetric generalized doubly stochastic inverse M -matrix problem for lists of real numbers and for lists of complex numbers of the form Λ = λ 1, a ± b i, …, a ± b i . The constructive nature of our results allows for the computation of a solution matrix. The paper also discusses an application of M -matrices to a capacity problem in wireless communications.