Let $B : \mathbb R^{m \times n} \to \mathbb R^{p \times p}$. Let $k_{ij}$ denote the $(i,j)$-th entry of matrix $K$. Is the following equation correct?
$$\text{Tr} \left(\frac{\partial B(K)}{\partial k_{ij}}\right)=\left(\frac{\partial \text{Tr}(B(K))}{\partial K}\right)_{ij}$$
Thanks for any ideas.
Thanks! @runway44
Summary $$ \begin{array}{rcl} \text{Tr} \left(\dfrac{\partial B(K)}{\partial k_{ij}}\right)&=&\text{Tr}\left(\displaystyle\lim_{\Delta k_{ij}\rightarrow 0}\dfrac{B(k_{ij}+\Delta k_{ij})-B(k_{ij})}{\Delta k_{ij}}\right)\\ &=&\displaystyle\lim_{\Delta k_{ij}\rightarrow 0} \text{Tr}\left(\dfrac{B(k_{ij}+\Delta k_{ij})-B(k_{ij})}{\Delta k_{ij}}\right)\quad \text{(Trace is continuous)}\\ &=&\displaystyle\lim_{\Delta k_{ij}\rightarrow 0}\dfrac{\text{Tr} (B(k_{ij}+\Delta k_{ij}))-\text{Tr}(B(k_{ij}))}{\Delta k_{ij}} \quad\text{(Trace is linear)}\\ &=&\dfrac{\partial \text{Tr}(B(K))}{\partial k_{ij}}\\ &=&\left(\dfrac{\partial \text{Tr}(B(K))}{\partial K}\right)_{ij} \end{array} $$