To find the eigenvectors $\mathbf X$ of a matrix $\mathbf A$, we construct the eigen equation

where $\lambda$ is the eigenvalue.

To make it seaier to understand, we rewrite it in the components form,

Mathematically speaking, it is straightforward to find the eigenvectors and eigenvalues.

Reconstruct $\mathbf A$

We can reconstruct $\mathbf A$ using the eigenvalues and eigenvectors.

First of all, we will construct a matrix of eigenvectors,

We also construct a diagonalized matrix $\mathbf \Lambda$

The original matrix $A$ is reconstrut by