Derivative of trace of product of matrices

Question

How to find the derivative of $$L=trace(A\left( \theta \right)^{-1}B\left( \theta \right))$$ wrt to $\theta \mathbf{}$. Where $A\left( \theta \right) $ and $B\left( \theta \right) $ are square matrices, and are functions of some vector of parameters $\theta \mathbf{.}$ Thanks a lot!

Answer 1

Hint: Consider your function $L:\mathbb{R}^n\longrightarrow\mathbb{R}$. Then if you want to find the derivative of $L$ at $\theta_0\in\mathbb{R}^n$ in direction $\hat\theta\in T_\theta\mathbb{R}^n\cong\mathbb{R}^n$, you can simply compute $$(D_\theta L)(\hat\theta) = \left.\frac{d}{dt}\right|_{t=0}L(\theta + t\hat\theta).$$

Answer 2

Recall that the derivative of ${\rm inv}\colon A \mapsto A^{-1}$ is given by $$ D(\def\i{{\rm inv}}\i)(A)H = -A^{-1}HA $$ Hence, by the chain and the product rule, we have, as ${\rm trace}$ is linear: $$ DL(\theta)(h) = {\rm trace}\bigl(A(\theta)^{-1}DB(\theta)h\bigr) - {\rm trace}\bigl(A(\theta)^{-1}DA(\theta)h \,A(\theta)^{-1}B(\theta)\bigr) $$