Article: Applications of the Generalized Fourier Transform in Numerical Linear Algebra
Author: K. Åhlander, H. Munthe-Kaas
Journal: BIT Numerical Mathematics (2005) 45: 819–850.
Abstract: Equivariant matrices, commuting with a group of permutation matrices, are con- sidered. Such matrices typically arise from PDEs and other computational problems where the computational domain exhibits discrete geometrical symmetries. In these cases, group representation theory provides a powerful tool for block diagonalizing the matrix via the Generalized Fourier Transform (GFT). This technique yields sub- stantial computational savings in problems such as solving linear systems, computing eigenvalues and computing analytic matrix functions such as the matrix exponential.
The paper is presenting a comprehensive self contained introduction to this field. Building upon the familiar special (finite commutative) case of circulant matrices be- ing diagonalized with the Discrete Fourier Transform, we generalize the classical con- volution theorem and diagonalization results to the noncommutative case of block diagonalizing equivariant matrices.
Applications of the GFT in problems with domain symmetries have been developed by several authors in a series of papers. In this paper we elaborate upon the results in these papers by emphasizing the connection between equivariant matrices, block group algebras and noncommutative convolutions. Furthermore, we describe the algebraic structure of projections related to non-free group actions. This approach highlights the role of the underlying mathematical structures, and provides insight useful both for software construction and numerical analysis. The theory is illustrated with a selection of numerical examples.
Remarks in hindsight: None yet.
1 Jul 2004