On the Inverse M-Matrix Problem for Real Symmetric Positive-Definite Toeplitz Matrices

Necessary and sufficient conditions are obtained for a real symmetric positive-definite Toeplitz matrix R to be an inverse of an M-matrix in terms of its Schur coefficients. Related problems are also considered, such as when such a matrix R can be extended to a higher-dimensional real symmetric positive-definite Toeplitz matrix whose inverse is an M-matrix or, under less restrictive conditions on R, when only its Cholesky factors are inverses of M-matrices. The proofs are constructive and allow the generation of such R’s with the various aforementioned properties.