Schur-type methods based on subspace considerations

Abstract

Generalizations of the Schur algorithm are presented and their relation and application to several algorithms in signal processing and linear algebra is elaborated. Based on an algebraic formulation, Schur's algorithm (for symmetric positive definite Toeplitz matrices) is generalized to more general matrices such as symmetric positive definite matrices, symmetric matrices, and general rectangular matrices. The resulting Schur-type methods are related to matrix decompositions such as Cholesky decomposition, R/sup T/DR-decomposition, and implicit Cholesky decomposition. When the number of hyperbolic rotations is minimized (which simultaneously maximizes the number of circular rotations) based on a subspace criteria, the relationship between the Schur algorithm and these decompositions as well as the suitability of the Schur algorithm for various signal processing applications (particularly signal/noise subspace estimation) becomes evident.