## Mathematics Publications

Mathematics

Article

#### Publication Version

Accepted Manuscript

9-15-2002

#### Journal or Book Title

Linear Algebra and its Applications

353

159

168

#### DOI

10.1016/S0024-3795(02)00301-4

#### Abstract

The symmetric M-matrix and symmetric M0-matrix completion problems are solved and results of Johnson and Smith [Linear Algebra Appl. 290 (1999) 193] are extended to solve the symmetric inverse M-matrix completion problem: (1) A pattern (i.e., a list of positions in an n×n matrix) has symmetric M-completion (i.e., every partial symmetric M-matrix specifying the pattern can be completed to a symmetric M-matrix) if and only if the principal subpattern R determined by its diagonal is permutation similar to a pattern that is block diagonal with each diagonal block complete, or, in graph theoretic terms, if and only if each component of the graph of R is a clique. (2) A pattern has symmetric M0-completion if and only if the pattern is permutation similar to a pattern that is block diagonal with each diagonal block either complete or omitting all diagonal positions, or, in graph theoretic terms, if and only if every principal subpattern corresponding to a component of the graph of the pattern either omits all diagonal positions, or includes all positions. (3) A pattern has symmetric inverse M-completion if and only if its graph is block-clique and no diagonal position is omitted that corresponds to a vertex in a graph-block of order >2. The techniques used are also applied to matrix completion problems for other classes of symmetric matrices.

This is a manuscript of an article published as Hogben, Leslie. "The symmetric M-matrix and symmetric inverse M-matrix completion problems." 353, no. 1-3 Linear Algebra and its Applications (2002): 159-168. DOI: 10.1016/S0024-3795(02)00301-4. Posted with permission.

#### Rights

Elsevier Science Inc.

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