How does one derive with respect to a matrix when that matrix is part of a summation whose result is inverted?

Question

A and B are matrices, a and b are scalars.

How would I derive the following expression with respect to A?

$$ \frac{\partial} {\partial A} ((a* A + b * B)^{-1})$$

Answer 1

Let $C=(aA+bB)$ and $F=C^{-1}$. Note that $I=CC^{-1}=CF$.

Take the differential of that last expression and simplify it $$\eqalign{ 0 &= C\,dF+dC\,F \cr C\,dF &= -dC\,F \cr dF &= -F\,dC\,F \cr }$$ Now express it in terms of $A$. $$\eqalign{ dF &= -Fa\,dA\,F \cr }$$ At this point, we're stuck because the derivative of a matrix wrt a matrix is a 4th order tensor.

One way to proceed is to vectorize the matrices by column stacking. $$\eqalign{ {\rm vec}(dF) &= -{\rm vec}(Fa\,dA\,F) \cr &= -(F^T\otimes Fa)\,{\rm vec}(dA) \cr \frac{\partial\,{\rm vec}(F)}{\partial\,{\rm vec}(A)} &= -F^T\otimes Fa \cr }$$ where $\otimes$ is the Kronecker product.