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.