On feedback equivalence and completion problems

AbstractIn 1987, I. Zaballa characterized the possible similarity classes of a square matrix with some prescribed rows. In 1988, the same author characterized the possible feedback equivalence classes of [A B], where A is a fixed square matrix and B varies. Firstly, in this paper, we observe that these results are equivalent, that is, each one of them can be obtained as a corollary of the other. Then, we apply similar arguments to other inverse problems. In particular, we study the possible invariant polynomials and the possible characteristic polynomials of A + BX + YC, when X and Y vary, and we study the linear systems that can become completely observable with a suitable choice of linear state feedback control