On a class of matrices with low displacement rank

Abstract

A matrix A such that, for some matrices U and V, the matrix AU−VA or the matrix A−VAU has a rank which is small compared with the order of the matrix is called a matrix with displacement structure. In this paper the authors single out a new class of matrices with displacement structure—namely, finite sections of recursive matrices—which includes the class of finite Hurwitz matrices, widely used in computer graphics. For such matrices it is possible to give an explicit evaluation of the displacement rank and an algebraic description, and hence a stable numerical evaluation, of the corresponding generators. The generalized Schur algorithm can therefore be used for the fast factorization of some classes of such matrices, and exhibits a backward stable behavior.