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

The power and Arnoldi methods in an algebra of circulants