One speaks normally of two main approaches to solving the eigenvalue problem.

• The first is the formal method, involving determinants and the characteristic polynomial. This proves how many eigenvalues there are, and is the way most of you learned about how to solve the eigenvalue problem, but for matrices of dimensions greater than 2 or 3, it is rather impractical.
• The other general approach is to use similarity or unitary tranformations to reduce a matrix to diagonal form. This is normally done in two steps: first reduce to for example a tridiagonal form, and then to diagonal form. The main algorithms we will discuss in detail, Jacobi's and Householder's (so-called direct method) and Lanczos algorithms (an iterative method), follow this methodology.