Reduction to Tridiagonal Form and Minimal Realizations

Abstract

This paper presents the theoretical background relevant to any method for producing a tridiagonal matrix similar to an arbitrary square matrix. Gragg’s work on factoring Hankel matrices and the Kalman–Gilbert structure theorem from systems theory both find a place in the development.Tridiagonalization is equivalent to the application of the generalized Gram–Schmidt process to a pair of Krylov sequences. In Euclidean space proper normalization allows one to monitor a tight lower bound on the condition number of the transformation. The various possibilities for breakdown find a natural classification by the ranks of certain matrices.The theory is illustrated by some small examples and some suggestions for restarting are evaluated This paper presents the theoretical background relevant to any method for producing a tridiagonal matrix similar to an arbitrary square matrix. Gragg’s work on factoring Hankel matrices and the Kalman–Gilbert structure theorem from systems theory both find a place in the development.Tridiagonalization is equivalent to the application of the generalized Gram–Schmidt process to a pair of Krylov sequences. In Euclidean space proper normalization allows one to monitor a tight lower bound on the condition number of the transformation. The various possibilities for breakdown find a natural classification by the ranks of certain matrices.The theory is illustrated by some small examples and some suggestions for restarting are evaluated