A new approach to square matrix completion problem

ABSTRACT

The classical completion problem of describing the possible similarity class of a square matrix with a prescribed arbitrary submatrix was studied by many authors through time, and it is completely solved in [Dodig M, Stošić M. Similarity class of a matrix with prescribed submatrix. Linear Multilinear Algebra. 2009;57:217–245; Combinatorics of polynomial chains. Linear Algebra Appl. 2020;589:130–157]. In this paper we show a surprising relation between this notable problem and the problem of describing the feedback invariants of restrictions and quotients of series connected systems studied in [Baragaña I, Zaballa I. Feedback invariants of restrictions and quotients: series connected systems. Linear Algebra Appl. 2002;351-352:69–89; Dodig M, Silva FC. Controllability of series connections of arbitrarily many linear systems. Linear Algebra Appl. 2008;429:122–141]. As a corollary, we obtain a new combinatorial result on partitions of integers.

KEYWORDS:

• Completion of matrix pencils
• partitions of integers
• classical majorization

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 05A17
• 15A83

Acknowledgments

The authors would like to thank the referee for comments and suggestions. This work was done within the activities of CEAFEL and was partially supported by FCT, projects UIDB/04721/2020, Exploratory Grants IF/01232/2014/CP1216/CT0012 (M.D.) and IF/0998/2015 (M.S.), and by the Ministry of Education, Science and Technological Development of the Republic of Serbia through Mathematical Institute SANU.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was done within the activities of CEAFEL and was partially supported by FCT, projects UIDB/04721/2020, Exploratory Grants IF/01232/2014/CP1216/CT0012 (M.D.) and IF/0998/2015 (M.S.), and by the Ministry of Education, Science and Technological Development of the Republic of Serbia through Mathematical Institute SANU.