The Bauer-type factorization of matrix polynomials revisited and extended

For a Laurent polynomial $a(\lambda)$, which is Hermitian and positive definite on the unit circle, the Bauer method provides the spectral factorization $a(\lambda)=p(\lambda)p^*(\lambda^{-1})$, where $p(\lambda)$ is a polynomial having all its roots outside the unit circle. Namely, as the size of the banded Hermitian positive definite Toeplitz matrix associated with the Laurent polynomial increases, the coefficients at the bottom of its Cholesky lower triangular factor tend to the coefficients of $p(\lambda)$. We study extensions of the Bauer method to the non-Hermitian matrix case. In the Hermitian case, we give new convergence bounds with computable coefficients.