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I'm trying to find a expression for the matrix derivative with respect to the pseudo-inverse of a matrix. So, i have some function $f(A)$ of a matrix $A$, which is singular. If it weren't I could use that $$ \frac{df(A)}{dA^{-1}} = -A^{-1}\frac{df(A)}{dA}A^{-1}, $$ but I can't right? So does anyone know where I could find a pseudo-inverse version of this? So basically I want an expression for $\frac{df(A)}{dA^+}$, and yes I reckon it won't be as cleas and simple as the one above. Also, does anyone know where I could find pseudo-inverse generalizations of all those classic matrix inversion lemmas?

Thanks in advance for any comments!

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The fonts render terribly, but there appears to be a derivation in the 'Mathematica Journal': mathematica-journal.com/issue/v8i4/inout/contents/… –  Steven Pav Jun 25 at 4:22
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1 Answer 1

To address your second question, here are two useful references:

  1. An Extension of the Matrix Inversion Lemma by Nariyasu Minamide in SIAM J. Alg. and Disc. Methods, 6, pp. 371-377 (1985).
  2. The Moore-Penrose generalized inverse for sums of matrices by J. A. Fill and D. E. Fishkind. (1998)
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Cool, thanks for the links. This seems to answer the second question pretty well! –  alexsuse Feb 7 '12 at 13:55
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