A unification of unitary similarity transforms to compressed representations

In this paper a new framework for transforming arbitrary matrices to compressed representations is presented. The framework provides a generic way of transforming a matrix via unitary similarity transformations to e.g.\ Hessenberg, Hessenberg\-/like and combinations of both. The new algorithms are deduced, based on the $QR$-factorization of the original matrix. Based on manipulations with Givens transformations, all the algorithms consists of eliminating the correct set of Givens transformations, resulting in a matrix obeying the desired structural constraints. Based on this new reduction procedure we investigate further correspondences such as irreducibility, unicity of the reduction procedure and the link with (rational) Krylov methods. The unitary similarity transform to Hessenberg\-/like form as presented here, differssignificantly from the one presented in earlier work. Not only does it use less Givens transformations to obtain the desired structure, also the convergence to rational Ritz values is not observed in the standard way