An Explicit Formula for Lanczos Polynomials*

The Lanczos algorithm for tridiagonalizing a given matrix A generates a sequence of approximating matrices A, that can naturally be obtained as restrictions to subspaces. The eigenvalues of these approximating matrices are well known to be good approximations to the extreme eigenvalues of A. In this paper we produce explicit formulas for the characteristic polynomials of the A,, in terms of the eigenvalues of A. These formulas can be used to explain heuristically why these approximations are often quite good. At present, we have no concrete analytic argument that explains the quality of the approximation. The main result draws on the formal relationship between the Lanczos algorithm and Pad& approximations to the moment generating function of A. This result is one of the few analytic results available for the unsymmetric Lanczos algorithm.