Applications of singular-value decomposition (SVD)

Abstract

Let A be an m x n matrix with m greater than or equal to n. Then one form of the singular-value decomposition of A is A = U-T SigmaV, where U and V are orthogonal and Sigma is square diagonal. That is, UUT = I-rank(A), VVT = I-rank(A), U is rank(A) x m, V is rank(A) x n and [GRAPHICS] is a rank (A) x rank(A) diagonal matrix. In addition sigma(1) greater than or equal to sigma(2) greater than or equal to... greater than or equal to sigma(rank)(A) > 0. The sigma(i)'s are called the singular values of A and their number is equal to the rank of A. The ratio sigma(1) /sigma(rank)(A) can be regarded as a condition number of the matrix A. It is easily verified that the singular-value decomposition can be also written as [GRAPHICS] The matrix u(i)(T) v(i) is the outerproduct of the i-th row of U with the corresponding row of V. Note that each of these matrices can be stored using only m + n locations rather than mn locations.