The power and Arnoldi methods in an algebra of circulants

TL;DR

This paper extends the power and Arnoldi methods to an algebra of circulants, enabling new computations in three-way data analysis with unique spectral properties and connections to Fourier-based techniques.

Contribution

It introduces the power and Arnoldi methods within a circulant algebra framework, defining new algebraic notions and exploring their implications for eigenvalues and algorithms.

Findings

Eigenvalues of circulant matrices are polynomial in dimension.

Eigenvalues can be represented by a canonical set.

Connections to Fourier transform techniques.

Abstract

Circulant matrices play a central role in a recently proposed formulation of three-way data computations. In this setting, a three-way table corresponds to a matrix where each "scalar" is a vector of parameters defining a circulant. This interpretation provides many generalizations of results from matrix or vector-space algebra. We derive the power and Arnoldi methods in this algebra. In the course of our derivation, we define inner products, norms, and other notions. These extensions are straightforward in an algebraic sense, but the implications are dramatically different from the standard matrix case. For example, a matrix of circulants has a polynomial number of eigenvalues in its dimension; although, these can all be represented by a carefully chosen canonical set of eigenvalues and vectors. These results and algorithms are closely related to standard decoupling techniques on block-circulant matrices using the fast Fourier transform.