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Positive, Path Product, and Inverse M-matrices

An M-matrix is a square matrix with nonnegative off-diagonal M-matrices and a nonnegative inverse. The nonnegative matrices that occur as inverses are called "inverse M-matrices" (IM for short). Of course, not any nonnegative matrix is inverse M. We consider the relations among entry-wise positive matrices, IM matrices and an intermediate class, the path-product matrices. It is shown that any positive matrix may be made path product via a sufficient addition to its diagonal and any path product product matrix may be made IM via a bounded addition to its diagonal. These results have implications for questions about Hadamard products of IM matrices.
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