Real Rectangular Matrices

Abstract

The FORTRAN codes in this Chapter address the question of computing distinct singular values and corresponding left and right singular vectors of real rectangular matrices, using a single-vector Lanczos procedure. For a given real rectangular ℓ × n matrix A, these codes compute nonnegative scalars σ and corresponding real vectors x ≠ 0 and y ≠ 0 such that EquationSource$$ % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGbbGaamiEaiabg2da9iabeo8aZjaadMhaieaacaWFGaGaamyy % aiaad6gacaWGKbGaa8hiaiaadgeapaWaaWbaaSqabeaapeGaamivaa % aakiaadMhacqGH9aqpcqaHdpWCcaWG4bGaaiOlaaaa!4716! $$Ax = \sigma y and {A^T}y = \sigma x.$$$$ ((6.1.1)) Every real rectangular ℓxn matrix, where ℓ ≥ n, has a singular value decomposition, EquationSource$$ % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGbbGaeyypa0Jaamywaiabfo6atjaadIfapaWaaWbaaSqabeaa % peGaamivaaaaieaak8aacaWFGaGaam4DaiaadMgacaWG0bGaamiAai % aa-bcapeGaamiwaiabg2da9iaadIfapaWaaWbaaSqabeaapeGaamiv % aaaak8aacaWGybGaeyypa0JaamysaiaacYcacaWFGaGaamywamaaCa % aaleqabaGaamivaaaakiaadMfacqGH9aqpcaWGjbGaa8hiaiaa-fga % caWFUbGaa8hzaiaa-bcacqqHJoWucaWF9aWaamWaaeaafaqabeGaba % aabaGaeu4Odm1aaSbaaSqaaiaadMeaaeqaaaGcbaGaaGimaaaaaiaa % wUfacaGLDbaaaaa!59A3! $$A = Y\Sigma {X^T} with X = {X^T}X = I, {Y^T}Y = I and \Sigma = \left[ {\matrix {{\Sigma _I}} \\ 0 \\ \endmatrix } \right]$$$$ ((6.1.2)) where Σ is ℓ × n and = diag {σ1,..., σn} with σi, 1 ≤ i ≤ n, the singular values of A. X is a n × n orthogonal matrix, Y is a ℓ × ℓ orthogonal matrix, and the columns of X and of Y are respectively, right and left singular vectors of A. There are many applications for this type of decomposition. Singular values and vectors are discussed in detail for example in Stewart [1973].KeywordsError EstimateSingular VectorMain ProgramVector ComputationReal Symmetric MatrixThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.