Eigendecomposition of A Matrix
Let A be a square (N×N) matrix with N linearly independent columns, Then A can be factorized as
where Q is the square (N×N) matrix whose ith column is the eigenvector of A and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, i.e., . Note that only diagonalizable matrices can be factorized in this way. For example, the defective matrix 
cannot be diagonalized.
The eigenvectors are usually normalized, but they need not be. A non-normalized set of eigenvectors, can also be used as the columns of Q. That this is true can be understood by noting that the magnitude of the eigenvectors in Q gets canceled in the decomposition by the presence of Q−1.
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