The Bauer-type factorization of matrix polynomials revisited and extended (Q1991634)

A matrix Laurent polynomial \( A(\lambda) = A_1\lambda + A_0 + A_{-1}\lambda^{-1}\) where \(A_1, A_0, A_{-1}\in\mathbb{C}^{m\times m}\) is called nonsingular if the polynomial \(\text{det}(\lambda A(\lambda))\) does not identically vanish. The authors consider only Laurent polynomials that admit a factorization \( A(\lambda) = (X\lambda + I)S(I +Y\lambda^{-1})\). The authors obtain formulas for the factors in the LDU factorization of the Toeplitz matrix associated with Laurent polynomials. They obtain convergence estimates for the Brauer-type factorization. They give a convergence analysis for a matrix Laurent polynomial which is Hermitian and positive semidefinite on the unit circle. They give applications to Brauer's method for the Laurent polynomials of high degree.