Applications of the Generalized Fourier Transform in Numerical Linear Algebra

Equivariant matrices, commuting with a group of permutation matrices, are considered. 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 substantial computational savings in problems such as solving linear systems, computing eigenvalues and computing analytic matrix functions such as the matrix exponential.