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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