It is well-known that while reducing a symmetric matrix into a similar tridiagonal one, the intermediate tridiagonal matrices contain the Lanczos-Ritz values as eigenvalues. Or for a Hessenberg matrix they contain the so-called Arnoldi-Ritz values. More information can be found in the following books [42,45,91,140,155,169] and the references therein.
In this section we provide necessary and sufficient conditions stating, whether partially reduced matrices during an orthogonal similarity reduction will contain the Arnoldi-Ritz values. It will also be shown that the reductions to a similar tridiagonal, Hessenberg, semiseparable and Hessenberg-like matrix are reductions satisfying the desired properties.