The Singular Value Decomposition and the Pseudoinverse
Gregor Gregorčič · 2001
where U is orthogonal m bym matrix and the columns of the U are the eigenvectors of AA . Likewise, V is orthogonal n by n matrix and the columns of the V are the eigenvectors of AA. The matrix S is diagonal and it is the same size as A. Its diagonal entries, also called sigma, σ1, . . . , σr, are the square roots of the nonzero eigenvalues of both AA and AA. They are the singular values of matrix A and they fill the first r places on the main diagonal of S. r is the rank of A. The connections with AA and AA must hold if the equation 1 is correct. It can be seen: AA = ( USV )( VSU ) = USSU (2)