On the Correlation Space of the Biproduct Coefficient Matrix Structure

The biproduct correlation coefficient matrix structure has gained acceptance and application in wireless communications. Yet, the set of correlation matrices that has this structure has not been fully identified. It is known that the positive equicorrelated matrix satisfies the biproduct condition, but little else is known. The set of admissible correlation matrices having the special biproduct structure is investigated. Constraints on the signs of the correlation coefficients that define the admissible set are revealed. Additional constraints on the magnitudes of the correlation coefficients for admissibilty are also identified. The set of matrices satisfying the required structure is much more restricted than previously thought. The constraints are shown to become increasingly restrictive as the dimension of the correlation matrix increases. The constraints also become more restrictive as the values of the correlation coefficients increase to approach 1. Previously, only the equicorrelated matrix has been identified as having the biproduct structure. Two additional correlation matrix structures satisfying the biproduct structure are identified. Tests that exclude a correlation matrix from having the biproduct structure are derived.