Calculating the Singular Values and Pseudo-Inverse of a Matrix

1965/01/01 by Gene H. Golub, W. Kahan · 9 citations
Computer Science · Physics and Astronomy · Mathematics · #Matrix Theory and Algorithms #Electromagnetic Scattering and Analysis #Scientific Research and Discoveries #Diagonal #Unitary matrix #Spurious relationship #Matrix (chemical analysis) #Mathematics #Inverse #Sigma #Unitary state #Diagonal matrix #Singular value #Cover (algebra) #Scheme (mathematics) #Applied mathematics #Algorithm #Mathematical analysis #Physics #Eigenvalues and eigenvectors #Geometry #Statistics

paper · doi:10.1137/0702016

openalex publication_date 1965/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/25

Abstract

A numerically stable and fairly fast scheme is described to compute the unitary matrices U and V which transform a given matrix A into a diagonal form Σ = U^ * AV, thus exhibiting A’s singular values on Σ ’s diagonal. The scheme first transforms A to a bidiagonal matrix J, then diagonalizes J. The scheme described here is complicated but does not suffer from the computational difficulties which occasionally afflict some previously known methods. Some applications are mentioned, in particular the use of the pseudo-inverse A^I = VΣ ^I U^* to solve least squares problems in a way which dampens spurious oscillation and cancellation.

Citations

• A generalized inverse for matrices
• The Approximation of One Matrix by Another of Lower Rank
• On some algorithms for the solution of the complete eigenvalue problem
• Unitary Triangularization of a Nonsymmetric Matrix

Cited by

• Accurate Singular Values of Bidiagonal Matrices
• Singular value decomposition and least squares solutions
• On the Early History of the Singular Value Decomposition
• Ill-Conditioned Eigensystems and the Computation of the Jordan Canonical Form
• On the Numerical Properties of an Iterative Method for Computing the Moore–Penrose Generalized Inverse
• On Iterative Computation of Generalized Inverses and Associated Projections
• Random matrix theory
• Eigenvalue computation in the 20th century
• Singular value decomposition [wikipedia]