Completions of M-matrix patterns Original Research Article

A list of positions in an n × n real matrix (a pattern) is said to haveM-completion if every partial M-matrix that specifies exactly these positions can be completed to an M-matrix. Let Q be a pattern that includes all diagonal entries and let G be its digraph. The following are equivalent. (1) the pattern Q has M-completion; (2) the pattern Q is permutation similar to a block triangular pattern with all the diagonal blocks completely specified; (3) any strongly connected subdigraph of G is complete. A pattern with some diagonal entries unspecified has M-completion if and only if the principal subpattern defined by the specified diagonal positions has M-completion.