The Uniform Correlation Matrix and its Application to Diversity

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, eigenvectors, and inverse. 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 semiuniform matrix, we obtain the eigenvalues, eigenvectors, and inverse of M(a, a*, L) in closed form. We present applications of these results to the analysis of communication systems using diversity under correlated fading conditions