Eigenvalues for Equivariant Matrices

An equivariant matrix A commutes with a group of permutation matrices. Such matrices often arise in numerical applications where the computational domain exhibits geometrical symmetries, for instance triangles, cubes, or icosahedra. The theory for block diagonalizing equivariant matrices via the Generalized Fourier Transform (GFT) is reviewed and applied to eigenvalue computations. For dense matrices which are equivariant under large symmetry groups, we give theoretical estimates that show a substantial performance gain. In case of cubic symmetry, the gain is about 800 times, which is verified by numerical results. It is also shown how the multiplicity of the eigenvalues is determined by the symmetry, which thereby restricts the number of distinct eigenvalues. The inverse GFT is used to compute the corresponding eigenvectors. It is emphasized that the inverse transform in this case is very fast, due to the sparseness of the eigenvectors in the transformed space.