Zeros of (Non-Square) Spectral Factors and Canonical Correlations

Abstract

Consider a left spectral factor of a rational spectral function. The system-zeros and zero directions of it cannot be arbitrary. They are determined by the zeros of the minim um phase spectral factor. Moreover, the partial ordering of the covariance matrices of the state vectors is reflected in the set of zeros and in the subspaces generated by the past of the state processes. Using this it is possible to give a probabilistic meaning to the eigenvalues of the matrix PQ, where P and Q are the covariance matrices of a forward and a backward realization, respectively. Namely, the square roots of these eigenvalues are the canonical correlations between the past of the forward and the future of the backwar realization.