Matrices with prescribed submatrices and number of invariant polynomials

AbstractWe give necessary and sufficient conditions for the existence of a matrix [A B], over a field, where A is a square matrix, such that [A B] has a prescribed submatrix and the pencil [xI - A - B] has a prescribed number of invariant factors different from 1. As a corollary we give a necessary and sufficient condition for the existence of a completely controllable pair (A, B) such that [A B] has a prescribed submatrix