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. © 2015 Elsevier Inc.