Suppose we have two real $n \times n$ matrices $A$ and $B$. Consider the linear operator $T = A^{-1} B$ which maps $\mathbb{R}^n$ to itself.
I'm trying to calculate a Taylor expansion of the function $T(A)$ about $A = B$,
$$
T(A) = I + \dfrac{\partial T}{\partial A}|_{A=B} (A – B) + \dfrac{1}{2} \dfrac{\partial^2 T}{\partial A^2}|_{A=B} (A – B)^2 + \dots
$$
but I'm stuck at calculating $\partial T / \partial A$.
I know that $\partial A^{-1} / \partial A = -A^{-2}$, and that $\partial T / \partial A^{-1} = B^T$. In what order do these derivatives need to be multiplied to obtain $\partial T / \partial A$?
I'm also not sure whether the order of multiplication in my Taylor series is correct. Any advice on this would be much appreciated!
Best Answer
Hint: For $ \frac{\partial T}{\partial A} $ you can use the fact that
$$ \partial (XY) = (\partial X)Y + X(\partial Y) $$
Now substitute $ X \mapsto A^{-1} $ and $ Y \mapsto B $.