Spectral Decomposition

Definition

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

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

Suppose that $A$ is a symmetric matrix. Then the $Q$ matrix consists on orthogonal eigenvectors. Then,

$$A=Q\Lambda Q^T$$

In that case $Q$ should have normalized eigenvectors since we do not eliminate such constants by taking the inverse.

Alternatively,

$$A_{k\times k}=\sum _{i=0}^n{\lambda }_i{e}_i{e}_i^T$$

where each $e_i$ is the normalized eigenvector of $A$.