Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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!

share|improve this question
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

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)
share|improve this answer
Cool, thanks for the links. This seems to answer the second question pretty well! –  alexsuse Feb 7 '12 at 13:55

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.