Campus Units
Mathematics
Document Type
Article
Publication Version
Accepted Manuscript
Publication Date
9-15-2002
Journal or Book Title
Linear Algebra and its Applications
Volume
353
First Page
159
Last Page
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.
Rights
This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Copyright Owner
Elsevier Science 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 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.