singular value decomposition

Any real m×n matrix A can be decomposed into


where U is an m×m orthogonal matrixMathworldPlanetmath, V is an n×n orthogonal matrix, and S is a unique m×n diagonal matrixMathworldPlanetmath with real, non-negative elements σi, i=1,,min(m,n) , in descending order:


The σi are the singular valuesMathworldPlanetmath of A and the first min(m,n) columns of U and V are the left and right (respectively) singular vectors of A. S has the form:


where Σ is a diagonal matrix with the diagonal elements σ1,σ2,,σmin(m,n). We assume now mn. If r=rank(A)<n , then


If σr0 and σr+1==σn=0, then r is the rank of A. In this case, S becomes an r×r matrix, and U and V shrink accordingly. SVD can thus be used for rank determination.

The SVD provides a numerically robust solution to the least-squares problem. The matrix-algebraic phrasing of the least-squares solution x is


Then utilizing the SVD by making the replacement A=USVT we have



  • Originally from The Data Analysis Briefbook (

Title singular value decomposition
Canonical name SingularValueDecomposition
Date of creation 2013-03-22 12:05:17
Last modified on 2013-03-22 12:05:17
Owner akrowne (2)
Last modified by akrowne (2)
Numerical id 9
Author akrowne (2)
Entry type Definition
Classification msc 15-00
Classification msc 65-00
Synonym SVD
Synonym singular value
Synonym singular vector
Related topic EigenvectorMathworldPlanetmathPlanetmathPlanetmath
Related topic EigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath