#### Campus Units

Mathematics

#### Document Type

Article

#### Publication Version

Submitted Manuscript

#### Publication Date

9-2002

#### Journal or Book Title

Linear Algebra and its Applications

#### Volume

353

#### DOI

10.1016/S0024-3795(02)00301-4

#### Abstract

The symmetric *M*-matrix and symmetric *M*0-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 *M*0-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.

#### Rights

This manuscript version is made available under the CCBY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/

#### Copyright Owner

Elsevier Inc.

#### Copyright Date

2002

#### Language

en

#### File Format

application/pdf

#### Recommended Citation

Hogben, Leslie, "The Symmetric M-Matrix and Symmetric Inverse M-Matrix Completion Problems" (2002). *Mathematics Publications*. 75.

https://lib.dr.iastate.edu/math_pubs/75

## Comments

This is a manuscript of an article from

Linear Algebra and its Applications353 (2002): 159, doi:10.1016/S0024-3795(02)00301-4. Posted with permission.