New Square-Root Factorization of Inverse Toeplitz Matrices

Square-root (in particular, Cholesky) factorization of Toeplitz matrices and of their inverses is a classical area of research. The Schur algorithm yields directly the Cholesky factorization of a symmetric Toeplitz matrix, whereas the Levinson algorithm does the same for the inverse matrix. The objective of this letter is to use results from the theory of rational orthonormal functions to derive square-root factorizations of the inverse of an ntimesn positive definite Toeplitz matrix. The main result is a new factorization based on the Takenaka-Malmquist functions, that is parameterized by the roots of the corresponding auto-regressive polynomial of order n . We will also discuss briefly the connection between our analysis and some classical results such as Schur polynomials and the Gohberg-Semencul inversion formula.