Can anyone help me simplify the following equation:
$\text{tr} \left [ \mathbf{A}^{-1} \left ( \mathbf{B} \circ \mathbf{A} \right ) \right ] = \text{tr} \left [ \mathbf{A}^{-1}\mathbf{C}\mathbf{A}^{-1}\left ( \mathbf{B} \circ \mathbf{A} \right ) \right ]$
The matrix $\mathbf{A}$ is parameterized as $A_{xy} = \text{exp}(d_{xy}/\ell)$ and I ultimately want to solve for $\ell$. My naive idea of simplifying using the invariance under circular permutations e.g. $\text{tr} \left [ \mathbf{A}^{-1} \left ( \mathbf{B} \circ \mathbf{A} \right ) \right ] = \text{tr} \left [ \left ( \mathbf{B} \circ \mathbf{A} \right ) \mathbf{A}^{-1} \right ] = \text{tr} \left [ \mathbf{B} \right ]$ does not hold since $\mathbf{A}\left(\mathbf{B} \circ \mathbf{C}\right) \neq \left(\mathbf{A}\mathbf{B}\right) \circ \mathbf{C}$.
The matrices $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ are all symmetric, which seems like it might help!
Any advice greatly appreciated!