An extended displacement operator for weakly structured covariance matrices

The Gohberg-Semencul formula is an explicit expression of the inverse of a Toeplitz matrix in terms of a reduced number of parameters which happen to be the forward and backward autoregressive parameters. It has been nicely understood in terms of displacement ranks. With the help of a new displacement operator, it is shown that this two-term formula remains valid in the general positive definite case, provided that the shifted predictors are now associated with the successive principal submatrices. Besides, this formula induces a general relationship among forward and backward predictors