Necessary and sufficient conditions for orthogonal similarity transformations to obtain the Arnoli(Lanczos)–Ritz values

It is a well-known fact that while reducing a symmetric matrix into a similar tridiagonal one, the already tridiagonal matrix in the partially reduced matrix has as eigenvalues the Lanczos–Ritz values. This behavior is also shared by the reduction algorithm which transforms symmetric matrices via orthogonal similarity transformations to semiseparable form. Moreover also the orthogonal reduction to Hessenberg form has a similar behavior with respect to the Arnoldi–Ritz values. In this paper we investigate the orthogonal similarity transformations creating this behavior. Two easy conditions are derived, which provide necessary and sufficient conditions, such that the partially reduced matrices have the desired convergence behavior. The conditions are easy to check as they demand that in every step of the reduction algorithm two particular matrices need to have a zero block.