Differentiating the trace with respect to a parameter

Question

Let us define a matrix $A(\rho)$ where $\rho \in \mathbb R$. Is there a formula for the derivative of $$\mbox{tr} \left( A(\rho)^{-1} \frac{\partial}{\partial \rho} A(\rho) \right)$$ with respect to $\rho$? Any kind of help would be much appreciated.

Answer 1

well, the trace of a matrix $M$ is \begin{equation} \text{tr}(M)=\sum_{i=1}^n a_{ii} \end{equation} so if $M$ is a differentiable function of a variable $\rho$, i guess we have \begin{equation} \frac{d}{d\rho}\text{tr}(M(\rho))=\sum_{i=1}^n a'_{ii}(\rho)=\text{tr}\left(\frac{d}{d\rho}M(\rho)\right) \end{equation}

Answer 2

Let $f : \mathbb R \to \mathbb R$ be defined by

$$f (t) := \mbox{tr} \left( \left(\mathrm A (t)\right)^{-1} \dot{\mathrm A} (t) \right)$$

Hence,

$$f (t + {\rm d} t) = \cdots = f (t) + \mbox{tr} \left( \left(\mathrm A (t)\right)^{-1} \ddot{\mathrm A} (t) \right) {\rm d} t - \mbox{tr} \left( \left(\mathrm A (t)\right)^{-1} \dot{\mathrm A} (t) \left(\mathrm A (t)\right)^{-1} \dot{\mathrm A} (t) \right) {\rm d} t$$

and, thus,

$$\dot f (t) = \mbox{tr} \left( \left(\mathrm A (t)\right)^{-1} \ddot{\mathrm A} (t) \right) - \mbox{tr} \left( \left(\mathrm A (t)\right)^{-1} \dot{\mathrm A} (t) \left(\mathrm A (t)\right)^{-1} \dot{\mathrm A} (t) \right)$$