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.