Fiedler-comrade and Fiedler--Chebyshev pencils

Fiedler pencils are a family of strong linearizations for polynomials expressed in the monomial basis, that include the classical Frobenius companion pencils as special cases. We generalize the definition of a Fiedler pencil from monomials to a larger class of orthogonal polynomial bases. In particular, we derive Fiedler-comrade pencils for two bases that are extremely important in practical applications: the Chebyshev polynomials of the first and second kind. The new approach allows one to construct linearizations having limited bandwidth: a Chebyshev analogue of the pentadiagonal Fiedler pencils in the monomial basis. Moreover, our theory allows for linearizations of square matrix polynomials expressed in the Chebyshev basis (and in other bases), regardless of whether the matrix polynomial is regular or singular, and for recovery formulas for eigenvectors, and minimal indices and bases

A note on generalized companion pencils in the monomial basis

In this paper, we introduce a new notion of generalized companion pencils for scalar polynomials over an arbitrary field expressed in the monomial basis. Our definition is quite general and extends the notions of companion pencil in De Terán et al. (Linear Algebra Appl 459:264&-333, 2014), generalized companion matrix in Garnett et al. (Linear Algebra Appl 498:360&-365, 2016), and Ma&-Zhan companion matrices in Ma and Zhan (Linear Algebra Appl 438: 621&-625, 2013), as well as the class of quasi-sparse companion pencils introduced in De Terán and Hernando (INdAM Series, Springer, Berlin, pp 157&-179, 2019). We analyze some algebraic properties of generalized companion pencils. We determine their Smith canonical form and we prove that they are all nonderogatory. In the last part of the work we will pay attention to the sparsity of these constructions. In particular, by imposing some natural conditions on its entries, we determine the smallest number of nonzero entries of a generalized companion pencilThis work has been partially supported by the Ministerio de Economía y Competitividad of Spain through Grants MTM2017-90682-REDT and MTM2015-65798-P

Duality of matrix pencils, Wong chains and linearizations

We consider two theoretical tools that have been introduced decades ago but whose usage is not widespread in modern literature on matrix pencils. One is dual pencils, a pair of pencils with the same regular part and related singular structures. They were introduced by V. Kublanovskaya in the 1980s. The other is Wong chains, families of subspaces, associated with (possibly singular) matrix pencils, that generalize Jordan chains. They were introduced by K.T. Wong in the 1970s. Together, dual pencils and Wong chains form a powerful theoretical framework to treat elegantly singular pencils in applications, especially in the context of linearizations of matrix polynomials. We first give a self-contained introduction to these two concepts, using modern language and extending them to a more general form; we describe the relation between them and show how they act on the Kronecker form of a pencil and on spectral and singular structures (eigenvalues, eigenvectors and minimal bases). Then we present several new applications of these results to more recent topics in matrix pencil theory, including: constraints on the minimal indices of singular Hamiltonian and symplectic pencils, new sufficient conditions under which pencils in L1, L2 linearization spaces are strong linearizations, a new perspective on Fiedler pencils, and a link between the Möller-Stetter theorem and some linearizations of matrix polynomials