  This paper examines the properties of real symmetric square matrices with a
constant value for the main diagonal elements and another constant value for
all off-diagonal elements. This matrix form is a simple subclass of circulant
matrices, which is a subclass of Toeplitz matrices. It encompasses other useful
matrices such as the centering matrix and the equicorrelation matrix, which
arise in statistical applications. We examine the general form of this class of
matrices and derive its eigendecomposition and other important properties. We
use this as a basis to look at the properties of the centering matrix and the
equicorrelation matrix, and various statistics that use these matrices.
