On a new kind of Ansatz Spaces for Matrix Polynomials

TL;DR

This paper introduces a new family of ansatz spaces for matrix pencils that generalize existing linearization techniques, ensuring almost all pencils in these spaces serve as strong linearizations for any matrix polynomial.

Contribution

It defines new ansatz spaces for matrix pencils, characterizes their properties, and demonstrates their broad applicability for linearizing matrix polynomials, including regular and singular cases.

Findings

Almost every matrix pencil in the new spaces is a strong linearization.

The intersection of multiple ansatz spaces is non-empty, enabling construction of large subspaces.

Original ansatz spaces can be derived from the new spaces via matrix transformations.

Abstract

In this paper, we introduce a new family of equations for matrix pencils that may be utilized for the construction of strong linearizations for any square or rectangluar matrix polynomial. We provide a comprehensive characterization of the resulting vector spaces and show that almost every matrix pencil therein is a strong lineariza- tion regardless whether the matrix polynomial under consideration is regular or sin- gular. These novel "ansatz spaces" cover all block-Kronecker pencils as a subset and therefore contain all Fiedler pencils modulo permutations. The important case of square matrix polynomials is examined in greater depth. We prove that the intersection of any number of block-Kronecker ansatz spaces is never empty and construct large subspaces of blocksymmetric matrix pencils among which still almost every pencil is a strong linearization. Moreover, we show that the original ansatz spaces L1 and L2 may essentially be recovered from block-Kronecker ansatz spaces via pre- and postmultiplication, respectively, of certain constant matrices.