Applications of singular-value decomposition (SVD)

Let A be an m×n matrix with m≥n. Then one form of the singular-value decomposition of A is A=UTΣV,where U and V are orthogonal and Σ is square diagonal. That is, UUT=Irank(A), VVT=Irank(A), U is rank(A)×m, V is rank(A)×n and Σ=σ10⋯000σ2⋯00⋮⋮⋱⋮⋮00⋯σrank(A)−1000⋯0σrank(A)is a rank(A)×rank(A) diagonal matrix. In addition σ1≥σ2≥⋯≥σrank(A)>0. The σi’s are called the singular values of A and their number is equal to the rank of A. The ratio σ1/σrank(A) can be regarded as a condition number of the matrix A.