Block minimal bases $\ell$-ifications of matrix polynomials
For researchers in numerical linear algebra and polynomial eigenvalue problems, this work provides a new theoretical framework for constructing low-degree matrix polynomials that can replace linearizations in certain applications, though the practical impact is incremental as it extends existing concepts.
This paper introduces a novel method for constructing strong ℓ-ifications of matrix polynomials, which are low-degree matrix polynomials preserving the eigenstructure of the original polynomial. The method works for ℓ < d when ℓ divides nd or md, and the resulting ℓ-ifications satisfy desirable properties such as strongness, uniform shifts of minimal indices, and recoverability of eigenvectors and minimal bases.
The standard way of solving a polynomial eigenvalue problem associated with a matrix polynomial starts by embedding the matrix coefficients of the polynomial into a matrix pencil, known as a strong linearization. This process transforms the problem into an equivalent generalized eigenvalue problem. However, there are some situations in which is more convenient to replace linearizations by other low degree matrix polynomials. This has motivated the idea of a strong $\ell$-ification of a matrix polynomial, which is a matrix polynomial of degree $\ell$ having the same finite and infinite elementary divisors, and the same numbers of left and right minimal indices as the original matrix polynomial. We present in this work a novel method for constructing strong $\ell$-ifications of matrix polynomials of size $m\times n$ and grade $d$ when $\ell< d$, and $\ell$ divides $nd$ or $md$. This method is based on a family called "strong block minimal bases matrix polynomials", and relies heavily on properties of dual minimal bases. We show how strong block minimal bases $\ell$-ifications can be constructed from the coefficients of a given matrix polynomial $P(λ)$. We also show that these $\ell$-ifications satisfy many desirable properties for numerical applications: they are strong $\ell$-ifications regardless of whether $P(λ)$ is regular or singular, the minimal indices of the $\ell$-ifications are related to those of $P(λ)$ via constant uniform shifts, and eigenvectors and minimal bases of $P(λ)$ can be recovered from those of any of the strong block minimal bases $\ell$-ifications. In the special case where $\ell$ divides $d$, we introduce a subfamily of strong block minimal bases matrix polynomials named "block Kronecker matrix polynomials", which is shown to be a fruitful source of companion $\ell$-ifications.