Constructing strong ℓ-ifications from dual minimal bases
Fernando De Terán, Froilán M. Dopico, Paul Van Dooren · Linear Algebra and its Applications · 2016
We provide an algorithm for constructing strong ℓ -ifications of a given matrix polynomial P ( λ ) of degree d and size m × n using only the coefficients of the polynomial and the solution of linear systems of equations. A strong ℓ -ification of P ( λ ) is a matrix polynomial of degree ℓ having the same finite and infinite elementary divisors, and the same numbers of left and right minimal indices as the original matrix polynomial P ( λ ) . All explicit constructions of strong ℓ -ifications introduced so far in the literature have been limited to the case where ℓ divides d , though recent results on the inverse eigenstructure problem for matrix polynomials show that more general constructions are possible. Based on recent developments on dual polynomial minimal bases, we present a general construction of strong ℓ -ifications for wider choices of the degree ℓ , namely, when ℓ divides one of nd or md (and d ≥ ℓ ). In the case where ℓ divides nd (respectively, md ), the strong ℓ -ifications we construct allow us to easily recover the minimal indices of P ( λ ) . In particular, we show that they preserve the left (resp., right) minimal indices of P ( λ ) , and the right (resp., left) minimal indices of the ℓ -ification are the ones of P ( λ ) increased by d − ℓ (each). Moreover, in the particular case ℓ divides d , the new method provides a companion ℓ -ification that resembles very much the companion ℓ -ifications already known in the literature.