The Lanczos Algorithm and Hankel Matrix Factorization

In 1950 Lanczos [22] proposed a method for computing the eigenvalues of symmetric and nonsymmetric matrices. The idea was to reduce the given matrix to a tridiagonal form, from which the eigenvalues could be determined. A characterization of the breakdowns in the Lanczos algorithm in terms of algebraic conditions of controllability and observability was addressed in [6] and [26]. Hankel matrices arise in various settings, ranging from system identification [23] to algorithmic fault tolerance [4]. In his 1977 dissertation, Kung [20] studied the Berlekamp-Massey (BM, 1967) algorithm [1], [24] for solving Hankel equations, and remarked that their algorithm is related to the Lanczos process. There still exists strong interest in a simple exposition of the BM algorithm; see, e.g., [19] in 1989. In 1971 Phillips [28] proposed a Hankel triangularization scheme, and presented a derivation of his method using a special symmetrized Lanczos process with a weighted and possibly indefinite inner product. In this paper, we present the first systematic treatment of the connections between the Lanczos process and the two Hankel algorithms. We show how the BM and Phillips algorithms are just special cases of the asymmetric Lanczos and symmetrized Lanczos algorithms, respectively, using particular choices for the matrix and starting vectors. In addition, we point