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By means of the singular value decomposition $ USV^*$ of an $ m\times n$ matrix $ A$, the solution of the minimization problem $ \Vert Ax-b\Vert _2\to\min$ can be expressed in the form

$\displaystyle x = A^+b,\quad A^+ = VS^+U^*,

where $ A^+$ is the so-called pseudo-inverse of $ A$ (Moore-Penrose inverse), and $ S^+$ is an $ n\times m$ diagonal matrix of the form

$\displaystyle S^+ =\operatorname{diag}(1/s_1,\ldots,1/s_k,0,\ldots,0)\, ,

containing the inverse values of the singular ones.


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  automatically generated 5/23/2011