  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.
