Real Rectangular Matrices
Jane K. Cullum, Ralph A. Willoughby · Birkhäuser Boston eBooks · 1985
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 (6.1.1) $$ % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGbbGaamiEaiabg2da9iabeo8aZjaadMhaieaacaWFGaGaamyy % aiaad6gacaWGKbGaa8hiaiaadgeapaWaaWbaaSqabeaapeGaamivaa % aakiaadMhacqGH9aqpcqaHdpWCcaWG4bGaaiOlaaaa!4716! $$Ax = \sigma y and {A^T}y = \sigma x.$$$$ Every real rectangular ℓxn matrix, where ℓ ≥ n, has a singular value decomposition, (6.1.2) $$ % 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! $$A = Y\Sigma {X^T} with X = {X^T}X = I, {Y^T}Y = I and \Sigma = \left[ {\matrix {{\Sigma _I}} \\ 0 \\ \endmatrix } \right]$$$$ 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].