[Math] Matrix derivative with respect to the pseudo-inverse

matrices

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!

Best Answer

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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