Properties of Zero-Free Spectral Matrices

In factor analysis, which is used for example in econometrics, by definition the number of latent variables has to exceed the number of factor variables. The associated transfer function matrix has more rows than columns, and when the factor variables are independent zero mean white noise sequences and the transfer function matrix is stable, then the output spectrum is singular. While a related paper focusses on the properties of such a nonsquare transfer function matrix, in this paper, we explore a number of properties of the spectral matrix and associated covariance sequence. In particular, a zero free minimum degree spectral factor can be computed with a finite number of rational calculations from the spectrum (in contrast to typical spectral factor calculations), assuming the spectrum fulfills a generic condition. Application of the result to Kalman filtering is indicated, and presentation of the results is also achieved using finite block Toeplitz matrices with entries obtained from the covariance of the latent variable vector.