Constructing strong ℓ-ifications from dual minimal bases

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.