On the characteristic polynomial of matrices with prescribed columns and the stabilization and observability of linear systems
Susana Furtado, Fernando C. Silva · Electronic Journal of Linear Algebra · 1998
Let A 2 F nn , B 2 F nt , where F is an arbitrary eld.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 de ned as the largest determinantal divisor (or the product of the invariant factors) of [ xIn , 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 elds a result, already proved for in nite elds, that describes all the possible characteristic polynomials of a square matrix when an arbitrary submatrix is xed and the other entries vary.Finally, applications to the stabilization and observability of linear systems by state feedback are studied.AMS subject classi cations.15A18, 93B60