On the number of invariant factors of partially prescribed matrices and control theory

Let A and B be matrices of sizes m x m and rn x LI, respectively, over an infinite field. Let i[A B] be the number of invariant factors of [xI - A / - B] different from 1. This paper gives a necessary and sufficient condition for the existence of a matrix [A B], with a prescribed submatrix on the right lower corner, such that i[A B] less than or equal to nu. In particular, this paper studies the possibility of a linear system; (chi)overdot (t) = A chi(t) + B zeta(t) being completely controllable, when a submatrix on the right lower corner of [A B] is unknown and all the other entries are prescribed. (C) 2000 Published by Elsevier Science Inc. All rights reserved.