## Involve: A Journal of Mathematics

• Involve
• Volume 10, Number 5 (2017), 781-799.

### Matrix completions for linear matrix equations

#### Abstract

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.

#### Article information

Source
Involve, Volume 10, Number 5 (2017), 781-799.

Dates
Revised: 14 June 2016
Accepted: 6 October 2016
First available in Project Euclid: 19 October 2017

https://projecteuclid.org/euclid.involve/1508433092

Digital Object Identifier
doi:10.2140/involve.2017.10.781

Mathematical Reviews number (MathSciNet)
MR3652447

Zentralblatt MATH identifier
1364.15021

Subjects
Primary: 15A83: Matrix completion problems
Secondary: 15A27: Commutativity

#### Citation

Buhl, Geoffrey; Cronk, Elijah; Moreno, Rosa; Morris, Kirsten; Pedroza, Dianne; Ryan, Jack. Matrix completions for linear matrix equations. Involve 10 (2017), no. 5, 781--799. doi:10.2140/involve.2017.10.781. https://projecteuclid.org/euclid.involve/1508433092

#### References

• M. Bakonyi and C. R. Johnson, “The Euclidean distance matrix completion problem”, SIAM J. Matrix Anal. Appl. 16:2 (1995), 646–654.
• L. M. DeAlba and L. Hogben, “Completions of P-matrix patterns”, Linear Algebra Appl. 319:1–3 (2000), 83–102.
• J. H. Drew, C. R. Johnson, S. J. Kilner, and A. M. McKay, “The cycle completable graphs for the completely positive and doubly nonnegative completion problems”, Linear Algebra Appl. 313:1–3 (2000), 141–154.
• R. Grone, C. R. Johnson, E. M. Sá, and H. Wolkowicz, “Positive definite completions of partial Hermitian matrices”, Linear Algebra Appl. 58 (1984), 109–124.
• L. Hogben, “Completions of inverse $M$-matrix patterns”, Linear Algebra Appl. 282:1–3 (1998), 145–160.
• L. Hogben, “Inverse $M$-matrix completions of patterns omitting some diagonal positions”, Linear Algebra Appl. 313:1–3 (2000), 173–192.
• L. Hogben, “Graph theoretic methods for matrix completion problems”, Linear Algebra Appl. 328:1–3 (2001), 161–202.
• R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge University Press, 1985. Reprinted in 1994.
• R. A. Horn and C. R. Johnson, Topics in matrix analysis, Cambridge University Press, 1991. Reprinted in 1994.
• C. R. Johnson and B. K. Kroschel, “The combinatorially symmetric $P$-matrix completion problem”, Electron. J. Linear Algebra 1 (1996), 59–63.
• C. R. Johnson and R. L. Smith, “The completion problem for $M$-matrices and inverse $M$-matrices”, Linear Algebra Appl. 241–243 (1996), 655–667.
• C. R. Johnson and Z. Wei, “Asymmetric TP and TN completion problems”, Linear Algebra Appl. 438:5 (2013), 2127–2135.
• K. Morris, “On the rank of a Kronecker sum of similar matrices”, capstone project, Georgia College & State University, 2015.
• H. Neudecker, “A note on Kronecker matrix products and matrix equation systems”, SIAM J. Appl. Math. 17:3 (1969), 603–606.
• E. Rukmangadachari, Mathematical methods, Dorling Kindersley, New Delhi, 2010.