Low rank perturbation of regular matrix polynomials

Abstract

Let A ( λ ) be a complex regular matrix polynomial of degree ℓ with g elementary divisors corresponding to the finite eigenvalue λ 0 . We show that for most complex matrix polynomials B ( λ ) with degree at most ℓ satisfying rank B ( λ 0 ) < g the perturbed polynomial ( A + B ) ( λ ) has exactly g - rank B ( λ 0 ) elementary divisors corresponding to λ 0 , and we determine their degrees. If rank B ( λ 0 ) + rank ( B ( λ ) - B ( λ 0 ) ) does not exceed the number of λ 0 -elementary divisors of A ( λ ) with degree greater than 1, then the λ 0 -elementary divisors of ( A + B ) ( λ ) are the g - rank B ( λ 0 ) - rank ( B ( λ ) - B ( λ 0 ) ) elementary divisors of A ( λ ) corresponding to λ 0 with smallest degree, together with rank ( B ( λ ) - B ( λ 0 ) ) linear λ 0 -elementary divisors. Otherwise, the degree of all the λ 0 -elementary divisors of ( A + B ) ( λ ) is one. This behavior happens for any matrix polynomial B ( λ ) except those in a proper algebraic submanifold in the set of matrix polynomials of degree at most ℓ . If A ( λ ) has an infinite eigenvalue, the corresponding result follows from considering the zero eigenvalue of the perturbed dual polynomial.