Some Properties of the Uniform Correlation Matrix and Their Applications

We consider a complex-valued L times L square matrix whose diagonal elements are unity, and lower and upper diagonal elements are the same, each lower diagonal element being equal to a (a ne 1) and each upper diagonal element being equal to b (b ne 1). We call this matrix the generalized semiuniform matrix, and denote it as M(a, b, L). For this matrix, we derive closed-form expressions for the characteristic polynomial, eigenvalues, and eigenvectors. Treating the non-real-valued uniform correlation matrix M(a, a^*,L), where (middot)^* denotes the complex conjugate and a ne a^*, as a Hermitian generalized semi-uniform matrix, we obtain the eigenvalues and eigenvectors of M(a, a^*, L) in closed form. The results are applied to the analysis of communication systems using diversity.