## Matrix Decomposition Theorem

Let be a Matrix of Eigenvectors of a given Matrix and a Matrix of the corresponding Eigenvalues. Then can be written

 (1)

where is a Diagonal Matrix and the columns of are Orthogonal Vectors. If is not a Square Matrix, then it cannot have a Matrix Inverse. However, if P is (with ), then can be written using a so-called Singular Value Decomposition of the form
 (2)

where and are Square Matrices with Orthogonal columns,
 (3)

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Singular Value Decomposition.'' §2.6 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 51-63, 1992.