On the Toeplitz embedding of an arbitrary matrix
In the first part of the paper the authors give a thorough analysis of the various polynomials associated with the inverse of the upper left principal minors of a block Toeplitz matrix. This provides a unified framework for deriving the basic formulae of the Levinson and the Trench algorithms for inverting block Toeplitz matrices; in case of ordinary Toeplitz matrices these algorithms require \(O(n^ 2)\) arithmetic operations (see e.g. the paper of \textit{B. Friedlander}, \textit{M. Morf}, \textit{I. Kailath} and \textit{L. Ljung} [ibid. 27, 31-60 (1979; Zbl 0414.15005)]). Another important consequence of the polynomial approach is the generalization of an earlier result of the authors on the Schur parametrization of positive definite block Toeplitz matrices [see SIAM J. Appl. Math. 36, 34-46 (1979; Zbl 0417.42013)]. In the second part the authors use the method of Delosme and Morf for the embedding of an arbitrary block matrix into a block Toeplitz matrix, and show that all results of the previous part can be carried over to arbitrary block matrices.
scientific article; zbMATH DE number 3855299
scientific article