Explicit block-structures for block-symmetric Fiedler-like pencils
For researchers in numerical linear algebra, this work offers a more explicit and canonical representation of block-symmetric linearizations, potentially aiding in the analysis of their numerical properties, though the contribution is incremental as it reorganizes existing families.
The paper introduces four families of block-symmetric pencils that serve as explicit block-structure representations for block-symmetric Fiedler-like pencils, showing that existing block-symmetric generalized Fiedler pencils (GFP) and generalized Fiedler pencils with repetition (GFPR) belong to these families after permutations. This provides an alternative, explicit approach to constructing structured linearizations of matrix polynomials.
In the last decade, there has been a continued effort to produce families of strong linearizations of a matrix polynomial $P(λ)$, regular and singular, with good properties. As a consequence of this research, families such as the family of Fiedler pencils, the family of generalized Fiedler pencils (GFP), the family of Fiedler pencils with repetition, and the family of generalized Fiedler pencils with repetition (GFPR) were constructed. In particular, one of the goals was to find in these families structured linearizations of structured matrix polynomials. For example, if a matrix polynomial $P(λ)$ is symmetric (Hermitian), it is convenient to use linearizations of $P(λ)$ that are also symmetric (Hermitian). Both the family of GFP and the family of GFPR contain block-symmetric linearizations of $P(λ)$, which are symmetric (Hermitian) when $P(λ)$ is. Now the objective is to determine which of those structured linearizations have the best numerical properties. The main obstacle for this study is the fact that these pencils are defined implicitly as products of so-called elementary matrices. In this paper we consider the family of block-minimal bases pencils, whose pencils are defined in terms of their block-structure, as a source of canonical forms for block-symmetric pencils. More precisely, we present four families of block-symmetric pencils which, under some generic nonsingularity conditions are block minimal bases pencils and strong linearizations of a matrix polynomial. We show that the block-symmetric GFP and GFPR, after some row and column permutations, belong to the union of these four families. Hence, these four families of pencils provide an alternative but explicit approach to the block-symmetric Fiedler-like pencils existing in the literature.