# Duality of matrix pencils and linearizations

Noferini, Vanni and Poloni, Federico (2013) Duality of matrix pencils and linearizations. [MIMS Preprint]

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 \emph{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 \emph{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 $\mathbb{L}_1$, $\mathbb{L}_2$ linearization spaces are strong linearizations, a new perspective on Fiedler pencils, and a link between the MÃ¶ller-Stetter theorem and some linearizations of matrix polynomials.