# Cards

## Diagnolize Matrice

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 $\mathbf A$.

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