Singular value decomposition and least squares solutions
Let A be a real mxn matrix with m>=n. It is well known (cf. [4] that $A U\sum {V^T} (1) where {U^T}U {V^T}V V{V^T} {I_n}\text{ and \sum \text{ diag(\sigma _{\text{1}\text{ \ldots \text{\sigma _n}\text{\text{ The matrix U consists of n orthonormalized eigenvectors associated with the n largest eigenvalues of AA T and the matrix V consists of the orthonormalized eigenvectors of A T A. The diagonal elements of are the non negative square roots of the eigenvalues of A T A; they are called singular values. We shall assume that \sigma _1} \geqq \sigma _2} \geqq \cdots \geqq \sigma _n} \geqq 0. Thus if rank(A)=r, s r+1 s r+2==s n 0. The decomposition (1) is called the singular value decomposition (SVD)
Milestones in Matrix Computation