ELA, Volume 4, pp. 19-31, August 1998, abstract.
On the Characteristic Polynomial of Matrices with Prescribed
Columns and the Stabilization and Observability of Linear Systems
Susana Furtado and Fernando C. Silva
Let A in F^{n times n}, B in F^{n times t}, where F is an arbitrary
field. In this paper, the possible characteristic polynomials of
[A, B], when some of its columns are prescribed and the other columns
vary, are described. The characteristic polynomial of [A, B] is
defined as the largest determinantal divisor (or the product of the
invariant factors) of [xI_n-A, -B]. This result generalizes a
previous theorem by H. Wimmer which studies the same problem when t=0.
As a consequence, it is extended to arbitrary fields a result, already
proved for infinite fields, that describes all the possible
characteristic polynomials of a square matrix when an arbitrary
submatrix is fixed and the other entries vary. Finally, applications
to the stabilization and observability of linear systems by state
feedback are studied.