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

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

where $\lambda$ is the eigenvalue.

We rewrite it in the components form,

$$$$A_{ij} x_j = \lambda x_i. \label{eqn-eigen-decomp-def}$$$$

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

## Eigenvectors are Special Directions

Judging from the definition in Eq.($\ref{eqn-eigen-decomp-def}$), the eigenvectors do not change direction under the operation of the matrix $\mathbf A$.

## Reconstruct $\mathbf A$

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

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

$$\mathbf P = \begin{pmatrix}\mathbf x_1 & \mathbf x_2 & \cdots & \mathbf x_n \end{pmatrix}.$$

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

$$\mathrm{diag} (\mathbf \Lambda) = \begin{pmatrix}\lambda_1 & \lambda_2 & \cdots & \lambda_n \end{pmatrix}.$$

The original matrix $A$ is reconstruct by

$$\mathbf A = \mathbf P \mathbf \Lambda \mathbf P^T.$$

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