Eigendecomposition of a Matrix

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Eigendecomposition of a Matrix

Definition (Eigendecomposition)

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

Find Eigenvalues:
1. For $\lambda_{i}, \ i=1,\cdots,n$ solve: $\det(A-\lambda_{i} I=0)$
Find Eigenvectors:
1. For each $\lambda_{i}, \ i=1,\cdots,n$ find eigenvector $v_{i}$ by solving the system of linear equations $(A-\lambda_{i} I)v_{i}=0$
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.
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.

Linked from

Principal Component Analysis