Authors: Dodig, Marija 
Stošić, Marko 
Title: Similarity class of a matrix with a prescribed submatrix
Journal: Linear and Multilinear Algebra
Volume: 57
Issue: 3
First page: 217
Last page: 245
Issue Date: 1-Jan-2009
Rank: M22
ISSN: 0308-1087
DOI: 10.1080/03081080801987726
Abstract: 
In this article we study the possible similarity classes of a square matrix when an arbitrary submatrix is prescribed. As the main result, we solve the even more general problem of describing the possible strict equivalence classes of a regular pencil when a subpencil is prescribed. This result improves the result by [Cabral and Silva, Similarity invariants of completions of submatrices, Lin. Alg. Appl. 169 (1992), 151-161.], since an explicit solution is obtained without any existential quantifiers involved, over an algebraically closed field. In fact, the sufficiency of the conditions obtained by [Gohberg, Kaashoek and Van Schangen, Eigenvalues of completions of submatrices, Lin. Multilin. Alg. 25 (1989), 55-70.] is proved. In the proof, we use various results and techniques including matrix pencils completions, Kronecker canonical form, Littlewood-Richardson coefficients, Young diagrams, the solution of the Carlson problem and localization techniques.
Keywords: Completions | Littlewood-Richardson coefficient | Localization | Matrix pencils
Publisher: Taylor & Francis
Project: FCT, Grant no. SFRH/ BPD/26607/2006

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