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Help me please to define Moore-Penrose pseudoinverse for square matrix.

Also, how can I use it in order to solve linear equations?

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Try en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse and if the information provided there should happen to be insufficient, please elaborate on what's missing or what you want to have clarified. –  t.b. Apr 13 '12 at 10:21

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The Moore-Penrose inverse of $A\in \mathbb{C}^{m\times n}$, denoted by $A^{+}$, is the unique matrix $X$ satisfying the following four Penrose equations. \begin{eqnarray} (i)~ AXA = A,~ (ii)~ XAX = X, ~ (iii)~ (AX)^* = AX,~ (iv)~ (XA)^* = XA \end{eqnarray}

There are various methods to find out Moore-Penrose inverse of a matrix. For example Rank factorization method, singular value decomposition method, QR decomposition etc. If $Ax = b$ is inconsistent then $x = A^{+}b$ represents the least square solution of minimum 2 norm.

In general $x = A^{+}b$ represents the least square solution to system $Ax = b$. For more detailed information refers to generalized inverse by Ben-Israel http://www.amazon.com/Generalized-inverses-applications-Adi-Ben-Israel/dp/0882759914

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