434 Linear Algebra
corollaryC.6 (The SVD theorem) LetA∈Rm,nwith rankr. ThenA=
UDV>whereDis anr×rmatrix with nonzero singular values ofAand the
columns ofU,V are orthonormal left and right singular vectors ofA. Further-
more, for alli,D^2 i,iis an eigenvalue ofA>A, theith column ofV is the cor-
responding eigenvector ofA>Aand theith column ofU is the corresponding
eigenvector ofAA>.