Reduction to Tridiagonal Form and Minimal Realizations

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