If λ_i 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:

${\displaystyle A=\sum _{i=1}^{n}\lambda _{i}v_{i}w_{i}^{T}}$

this is known as the spectral decomposition of A.