CIRCULATIVE MATRICES OF DEGREE THETA

In this paper a special class of matrices W in C(n x n) that are a generalization of reflexive and antireflexive matrices are introduced, their fundamental properties are developed, and a decomposition method associated with W is presented. The matrices W have the relation W = e(itheta) P*WP, i = square-root -1, theta is-an-element-of R, where e is the exponential function and P an n x n unitary matrix with the property P(k) = I, k greater-than-or-equal-to 1. The superscript * denotes the conjugate transpose and I is the identity matrix. It is assumed that k is finite and is the smallest positive integer for which the relation holds. The matrices W are referred to as circulative matrices of degree theta with respect to P. Embedded in this class of matrices are two special types of matrices U and V, U = P*UP and V = -P*VP, which bear a great resemblance to reflexive matrices and antireflexive matrices, respectively. The matrices U and V are simply called circulative matrices and anticirculative matrices, respectively, without referring to their degrees, for the sake of brevity. These matrices are introduced and general theories associated with them are developed. Then their special cases are discussed, and, in particular, two more special classes of matrices are defined, which will be referred to as rotative and antirotative matrices. These are a special case of the circulative/anticirculative matrices on one band and a generalization of (block) circulant/anticirculant matrices on the other. Numerical examples are presented.