Direct Block Tridiagonalization of Single-Input Single-Output Systems

: In this paper we derive a direct method for block tridiagonalizing a single-input single-output system triple fA; b; cg. The method is connected to the nonsymmetric Lanczos procedure developed in [10][2][1] and also leads to canonical representations of such triples. Keywords: Lanczos procedure, state space, canonical form, numerical algorithms, direct elimination. 1 Introduction The Lanczos method is a recursive process for tridiagonalizing a given real or complex matrix A via a similarity transformation T . In numerical linear algebra this is typically used for the computation of the eigenvalues of A by exploiting the resulting tridiagonal form. The recurrence of the unsymmetric Lanczos procedure [12] starts with two (arbitrary) vectors b and c and consists of a biorthogonalization of the Krylov sequences C n \Delta = h b; Ab; A 2 b; : : : ; A n\Gamma1 b i ; O n \Delta = 2 6 6 6 6 6 6 6 4 c cA cA 2 . . . cA n\Gamma1 3 7 7 7 7 7 7 7 5 ; (1) where n is the order o..