The completion problem for M-matrices and inverse M-matrices

The completion problems for M-matrices and inverse hi-matrices are discussed. The former has a simple solution, but is not at all analogous to the positive definite completion problem. The latter is more subtle and is analogous to the positive definite problem. A graph-theoretic completion result, as well as a determinant maximizing completion that is characterized by having a zero entry in the inverse in every position in which the partial matrix has an unspecified entry, is given for the inverse M-matrix problem. These mirror, to some extent, the positive definite case, but the methodology is necessarily different.