Mathematics-Online lexicon: Pseudo-Inverse

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	Mathematics-Online lexicon:
	Pseudo-Inverse

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By means of the singular value decomposition

of an $m\times n$ matrix

, 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

is the so-called pseudo-inverse of

(Moore-Penrose inverse), and

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.

Annotation:

automatically generated 5/23/2011