# Eigendecomposition

06-27-2022 || 01:48
Tags: #linear-algebra

# eigendecomposition

Eigendecomposition is used to decompose a matrix to different parts.

Like any integer can be decomposed into prime factors and that would give us more information on the integer itself like is it divisible by any number or multiplier of any number.

In the same way, matrics decomposition will give us more information on the matrix itself and will make easier for many functions to work on it.

Now how to find eigendecomposition,

1. We will first find the eigen value and vector using the formula of eigenvalue-eigenvector
2. Lets say, for a matrix $A$, we have eigenvalues $\lambda_1$, $\lambda_2$ and the corresponding eigenvectors as $v_1$, $v_2$
3. By formula we know that,
$$A v_1 = \lambda_1 v_1$$
$$A v_2 = \lambda_2 v_2$$
4. We will use these numbers in a matrix form by using the $v_1$, $v_2$ vectors in column
$$A \begin{vmatrix} v_{11} & v_{21} \\ v_{12} & v_{22} \end{vmatrix} = \begin{vmatrix} v_{11} & v_{21} \\ v_{12} & v_{22} \end{vmatrix} \begin{vmatrix} \lambda_{1} & 0 \\ 0 & \lambda_{2} \end{vmatrix}$$
5. If we simplify this by providing $V$ as the column form matrix and the $\lambda$ matrix as $\Uplambda$, then
$$A V = V \Uplambda$$
$$A = V \Uplambda V^{-1}$$
This is the eigendecomposition of matrix A

[!warning]
This can only be used for square matrices