       Re: What's behind PseudoInverse

• To: mathgroup at smc.vnet.net
• Subject: [mg4332] Re: What's behind PseudoInverse
• From: richard at seuss.math.wright.edu (Richard Mercer)
• Date: Thu, 11 Jul 1996 00:57:43 -0400
• Organization: Wright State University
• Sender: owner-wri-mathgroup at wolfram.com

```In <4q5kak\$bsb at dragonfly.wolfram.com> Lorenzo Flueckiger wrote:
> Hello,
>
> The Mathematica function "PseudoInverse" do exactly what I need (for
> inversion of rectangular numerical matrices)... But how it really works?
> Does anybody know where I can find some documentation (method used or
> algorithm or equivalent package in C or ...) about this "magic"
> function?
>
> Thanks in advance
>
> Lorenzo
>

There is a short but good discussion of pseudoinverses in Gil Strang's
"Introduction to Linear Algebra".
I suggest that you look carefully at it or similar account of this topic;
I do not think it makes sense to use pseudoinverses without understanding
them.

Very briefly, it is based on the "singular value decomposition" of a matrix,
which is a generalization to arbitrary matrices of the diagonalization of a
symmetric square matrix.
One uses the square roots of the eigenvalues of A*A to create a diagonal
matrix S, and orthogonal matrices U and V to write A = USV' (where the
apostrophe means transpose). Replacing S by its easily found inverse then
results in the pseudoinverse of A.

--
Richard Mercer
richard at seuss.math.wright.edu
"I meant what I said and I said what I meant,
An elephant's faithful, one hundred per cent."

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