On completion problems for various classes of P-matrices

Date
2006-03-01
Authors
Bowers, John
Evers, Job
Hogben, Leslie
Hogben, Leslie
Shaner, Steve
Snider, Karyn
Wangsness, Amy
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Altmetrics
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Research Projects
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Computer Science
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Mathematics
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Abstract

A P-matrix is a real square matrix having every principal minor positive, and a Fischer matrix is a P-matrix that satisfies Fischer’s inequality for all principal submatrices. In this paper, all patterns of positions for n × n matrices, n ⩽ 4, are classified as to whether or not every partial Π-matrix can be completed to a Π-matrix for Π any of the classes positive P-, nonnegative P-, or Fischer matrices. Also, all symmetric patterns for 5 × 5 matrices are classified as to completion of partial Fischer matrices, and all but two such patterns are classified as to positive P- or nonnegative P-completion. We also show that any pattern whose digraph contains a minimally chordal symmetric-Hamiltonian induced subdigraph does not have Π-completion for Π any of the classes positive P-, nonnegative P-, Fischer matrices.

Description
<p>This is a manuscript of an article from <em>Linear Algebra and its Applications </em>413 (2006): 342, doi:<a href="http://dx.doi.org/10.1016/j.laa.2005.10.007" target="_blank">10.1016/j.laa.2005.10.007</a>. Posted with permission.</p>
Keywords
Matrix completion, Partial matrix, P-matrix, Nonnegative P-matrix, Positive P-matrix, Fischer matrix, Weakly sign-symmetric P-matrix
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