Given a matrix $\mathbf A$, it is diagonalized using its eigenvectors.

Why are the eigenvectors needed?

1. Find the eigenvectors $\mathbf x_i$ of the matrix $\mathbf A$; If we find degerations, the matrix is not diagonalizable.
2. Construct a matrix $\mathbf S = \begin{pmatrix} \mathbf x_1 & \mathbf x_2 & \cdots & \mathbf x_n \end{pmatrix}$;
3. The matrix $\mathbf A$ is diagonalize using $\mathbf S^{-1} \mathbf A \mathbf S = \mathbf {A_D}$