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

Why are the eigenvectors needed?

Eigenvectors of a matrix $\mathbf A$ are the preferred directions. From the definition of eigenvectors,

$$\mathbf A \mathbf x = \lambda \mathbf x,$$

we know that the matrix $\mathbf A$ only scales the eigenvectors and no rotations. These directions are special to the matrix $\mathbf A$.

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 $\mathbf S^{-1} \mathbf A \mathbf S = \mathbf {A_D}$

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