The basic philosophy is to
- Either apply subsequent similarity transformations (direct method) so that
$$
\begin{equation}
\mathbf{S}_N^T\dots \mathbf{S}_1^T\mathbf{A}\mathbf{S}_1\dots \mathbf{S}_N=\mathbf{D} ,
\tag{2}
\end{equation}
$$
- Or apply subsequent similarity transformations so that \( \mathbf{A} \) becomes tridiagonal (Householder) or upper/lower triangular (the QR method to be discussed later).
- Thereafter, techniques for obtaining eigenvalues from tridiagonal matrices can be used.
- Or use so-called power methods
- Or use iterative methods (Krylov, Lanczos, Arnoldi). These methods are popular for huge matrix problems.