Eigenvalues and Eigenvectors

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,

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

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

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