Vol. 10, No. 5, 2017

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Matrix completions for linear matrix equations

Geoffrey Buhl, Elijah Cronk, Rosa Moreno, Kirsten Morris, Dianne Pedroza and Jack Ryan

Vol. 10 (2017), No. 5, 781–799

A matrix completion problem asks whether a partial matrix composed of specified and unspecified entries can be completed to satisfy a given property. This work focuses on determining which patterns of specified and unspecified entries correspond to partial matrices that can be completed to solve three different matrix equations. We approach this problem with two techniques: converting the matrix equations into linear equations and examining bases for the solution spaces of the matrix equations. We determine whether a particular pattern can be written as a linear combination of the basis elements. This work classifies patterns as admissible or inadmissible based on the ability of their corresponding partial matrices to be completed to satisfy the matrix equation. Our results present a partial or complete characterization of the admissibility of patterns for three homogeneous linear matrix equations.

matrix completion problems, partial matrices, matrix commutativity, matrix equations
Mathematical Subject Classification 2010
Primary: 15A83
Secondary: 15A27
Received: 22 November 2015
Revised: 14 June 2016
Accepted: 6 October 2016
Published: 14 May 2017

Communicated by Chi-Kwong Li
Geoffrey Buhl
Department of Mathematics
CA State Univ Channel Islands
1 University Dr.
Camarillo, CA 93012
United States
Elijah Cronk
Department of Mathematics
Ithaca College
953 Danby Rd.
Ithaca, NY 14850
United States
Rosa Moreno
Department of Mathematics
California State University Channel Islands
1 University Dr.
Camarillo, CA 93012
United States
Kirsten Morris
Department of Mathematics
The University of Georgia
University of Georgia
Athens, GA 30602
United States
Dianne Pedroza
Department of Mathematics
Ripon College
300 Seward St.
Ripon, WI 54971
United States
Jack Ryan
Department of Mathematics
The University of Tennessee Knoxville
1403 Circle Dr.
Knoxville, TN 37996
United States