Eigendecomposition

Definition

Eigendecomposition is factorization of a matrix into the canonical form, where it is written by using it’s eigenvectors and eigenvalues.

Spectral decomposition

If the factorized matrix $A$ is a normal or a symmetric matrix, then the related decomposition if called Spectral Decomposition.

Let $A$ be square $n\times n$ matrix with $n$ linearly independent eigenvectors $e_i$. Then it can be factored as:

$$Av=\lambda v$$ $$AQ=Q\Lambda$$ $$A=Q\Lambda Q^{-1}$$

Note that only diagonalizable matrices can be factorized in this way.