On the block lu decomposition of block toeplitz matrices using the vector yule-walker equations

Abstract

It is shown that the inverse of a block Toeplitz matrix can be factored into a product of an upper block triangular, a block diagonal and a lower block triangular matrices, where the component matrices consist of the solutions of the forward and backward extended Yule-Walker equations. Also, a recursive algorithm is presented to decompose nested Toeplitz matrices, which is a generalization of the Levinson-Durbin algorithm. Its derivation is elementary. The decomposition and the algorithm are useful for solving a block Toeplitz system of simultaneous equations, particularly, which appears in the vector autoregressive moving-average model analysis.