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Re: What's behind PseudoInverse

  • To: mathgroup at
  • Subject: [mg4332] Re: What's behind PseudoInverse
  • From: richard at (Richard Mercer)
  • Date: Thu, 11 Jul 1996 00:57:43 -0400
  • Organization: Wright State University
  • Sender: owner-wri-mathgroup at

In <4q5kak$bsb at> 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 

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
"I meant what I said and I said what I meant,
An elephant's faithful, one hundred per cent."


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