The Singular Value Decomposition and Least Squares Problems

From subsection (B) it is clear that some systems of linear equations Ax = b have no solution. This may occur for systems with the same number of unknowns as there are equations or for systems Ax = b with nonsquare system matrices A. There is a remedy for this, namely to try to find the least squares solution xls that minimizes the error 

To do so, we introduce the singular value decomposition of a real matrix Arn/n, 

Since the length of a vector is not altered under an orthogonal transformation (see subsection (F)), the SVD of A yields 

Note that no variation occurs in the second term (ufb)2 in the above for- 

This is the least squares solution for Ax = b because the th component of V1 xis is on = (ujb)/ai by the orthonormality of the columns vt of V so that the first sum term above becomes zero for xis- 

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G) The Singular Value Decomposition and Least Squares Problems... 

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