Circulative Matrices of Degree $\theta $

In this paper a special class of matrices W in $\mathcal{C}^{n \times 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 $W = e^{i\theta } P^ * WP,\,i = \sqrt { - 1} ,\theta \in \mathcal{R}$, where e is the exponential function and P an $n \times n$ unitary matrix with the property $P^k = I,\,k \geq 1$. 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 matrice...