The power and Arnoldi methods in an algebra of circulants

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. These results and algorithms are closely related to standard decoupling techniques on block-circulant matrices using the fast Fourier transform. 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, the number of eigenpairs may exceed the dimension of the matrix, though it is still polynomial in it. It is thus necessary to take an extra step and carefully select a smaller, canonical set of size equal to the dimension of the matrix from which all possible eigenpairs can be formed. Copyright c © 0000 John Wiley & Sons, Ltd.