Eigen Values and Eigen Vectors
Basic Definitions
\[\begin{gather*} Ax = \lambda x \end{gather*}\]- $x$ is an eigen vector, $\lambda$ is an eigen value
Important properties
- Only square matrices have eigen values and vectors
Eigen value Decomposition
Say a matrix $A$ has two eigen vectors $v_1$, $v_2$, and their corresponding eigen values are: $\sigma_1$, $\sigma_2$
Then, we have
\[\begin{gather*} A \begin{bmatrix} v_1 & v_2 \end{bmatrix} = \begin{bmatrix} v_1 & v_2 \end{bmatrix} \begin{bmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{bmatrix} \end{gather*}\]So, we can get Eigen Value Decomposition:
\[\begin{gather*} V = \begin{bmatrix} v_1 & v_2 \end{bmatrix}, \Lambda = \begin{bmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{bmatrix} \\ => A = V \Lambda V^{-1} \end{gather*}\]Applications
Series of self-multiplications
Assume we want to apply the same linear transform 8 times. Say, $A^8$
Matrix multiplication is expensive. One can use divide and conquer, and do the multiplication in the order of $log2(8)$ times.
But with Eigen Value Decomposition, this problem becomes:
\[\begin{gather*} A^8 = V \Lambda^8 V^{-1} \end{gather*}\]$\Lambda^8$ is easy to calculate, because it’s just a diagonal matrix.
