Engineering Analysis/Spectral Decomposition

If λ_i are are the n distinct eigenvalues of matrix A, and v_i are the corresponding n distinct eigenvectors, and if w_i are the n distinct left-eigenvectors, then the matrix A can be represented as a sum:

$A=\sum _{i=1}^n{\lambda }_i{v}_i{w}_i^T$

this is known as the spectral decomposition of A.