QR Method

The QR method can be used for arbitrary matrices, but becomes too laborious and instead it is used on special matrices, preferably Hessenberg or symmetric band matrices. The steps involved in its implementation [3] can be briefly summarized as follows:
1. QR decomposition
For a regular square matrix ${\bf A}$, there exists a decomposition  

$${\bf A} = {\bf Q} {\bf R},$$ (252)

where ${\bf Q}$ is a unitary matrix and ${\bf R}$ is an upper triangular matrix.
2. QR transformation
The QR decomposition is followed by a QR transformation  

$$\tilde{{\bf A}} = {\bf R} {\bf Q} = {\bf Q}^T {\bf A} {\bf Q}$$ (253)

3. QR algorithm This is an iterative algorithm

   $$\begin{eqnarray} {\bf A}_1 &\equiv& {\bf A} \nonumber \\ {\bf A}_k &=& {\bf Q}_k... ...R}_k {\bf Q}_k \nonumber \\ &=& {\bf Q}^T_k {\bf A}_k {\bf Q}_k .\end{eqnarray}$$

This iteration converges for the non-symmetric case to an upper block triangular matrix, which has single elements or $2 \times 2$blocks on the diagonal. Then the eigenvalues can be taken from the diagonal or by solving $2 \times 2$ eigenvalue problems. For the symmetric case, the limit matrix is a diagonal matrix.