Definition

Eigendecomposition is the process of a decomposing a square matrix into a set of eigenvectors and eigenvalues.

Theorem

Let $A$ be an $n\times n$ square matrix. If $A$ has $n$ linearly independent eigenvectors $\{v_{1},\cdots,v_{n}\}$, with corresponding eigenvalues $\{\lambda_{1},\cdots,\lambda_{n}\}$, then $A$ can be factorized as $$A=VDV^{-1}$$where $V$ is the matrix whose $i$-th column is the eigenvector $v_{i}$ of $A$, and $D$ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, i.e. $D=diag(\lambda_{1},\cdots,\lambda_{n})$ ## Steps for Eigendecomposition 1. Find Eigenvalues: 1. For $\lambda_{i}, \ i=1,\cdots,n$ solve: $$\det(A-\lambda_{i} I=0)$$ 2. Find Eigenvectors: 1. For each $\lambda_{i}, \ i=1,\cdots,n$ find eigenvector $v_{i}$ by solving the system of linear equations $$(A-\lambda_{i} I)v_{i}=0$$ 3. Construct the Matrix $V$ and $D$: 1. Form $V$ by placing eigenvectors as columns in $V$ 2. Form $D$ by placing eigenvalues on the diagonal. 4. Verification: 1. Verify the decomposition by checking if $A=VDV^{-1}$ holds.

Note

This method only works if $A$ is diagonalizable, if not then Jordan Canonical Form must be used.