Computation of the Singular Value Decomposition

then σ is a singular value of A and u and v are corresponding left and right singular vectors, respectively. (For generality it is assumed that the matrices here are complex, although given these results, the analogs for real matrices are obvious.) If, for a given positive singular value, there are exactly t linearly independent corresponding right singular vectors and t linearly independent corresponding left singular vectors, the singular value has multiplicity t and the space spanned by the right (left) singular vectors is the corresponding right (left) singular space. Given a complex matrix A having m rows and n columns, the matrix product U V∗ is a singular value decomposition for a given matrix A if U and V , respectively, have orthonormal columns. has nonnegative elements on its principal diagonal and zeros elsewhere. A = U V∗. Let p and q be the number of rows and columns of . U is m × p, p ≤ m, and V is n × q with q ≤ n. There are three standard forms of the SVD. All have the i th diagonal value of denoted σi and ordered as follows: σ1 ≥ σ2 ≥ · · · ≥ σk , and r is the index such that σr > 0 and either k = r or σr+1 = 0. 1. p = m and q = n. The matrix is m × n and has the same dimensions as A. 2. p = q = min{m, n}.The matrix is square. 3. If p = q = r, the matrix is square. This form is called a reduced SVD and denoted by Û ̂V̂∗.