Lie Algebraic Generalizations of Hessenberg Matrices and the Topology of Hessenberg Varieties

Abstract

Various problems in control and signal processing generate matrix eigenvalue problems for which the matrix in question belongs to a linear Lie algebra. For example, the problem of obtaining the stabilizing solution to the algebraic Riccati can be solved by computing an invariant subspace of a Hamiltonian matrix-i.e., a matrix belonging to the Lie algebra of the symplectic group. Since Hessenberg form is important for the efficient implementation of the QR-algorithm to solve matrix eigenvalue problems, it is of interest to understand how the notion of Hessenberg matrix-or more generally the notion of banded matrices-generalizes to an arbitrary (semisimple) Lie algebra. Closely related to Hessenberg matrices are certain nested sequences of subspaces called Hessenberg flags. Under certain conditions, Hessenberg flags correspond to symmetric banded (e.g., tridiagonal) matrices of fixed spectrum. We describe a natural generalization of Hessenberg matrices to an arbitrary semisimple Lie algebra. For each of the classical complex simple Lie algebras, we examine the topology of the associated varieties of Hessenberg flags. The structure of these varieties is related to the combinatorics of the height function on the root system of the Lie algebra.KeywordsBetti NumberEulerian NumberBorel SubalgebraFlag ManifoldHessenberg MatrixThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.