Similarity class of a matrix with a prescribed submatrix

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.