I was trying to differentiate the equation below: $$ \frac{\partial a^T X^{-T}X^{-1}a} {\partial X} $$ where X is invertible but not symmetric and $X^{-T}$ means transpose of inverse of X.
In the above when there is no $X^{-T}$ then it can be done from math cookbook. Can anybody help me to solve that please?
By the product rule \begin{align*} \frac{\partial}{\partial x_{ij}} &= a^T\frac{\partial X^{-T}}{\partial x_{ij}}X^{-1}a + a^TX^{-T}\frac{\partial X^{-1}}{\partial x_{ij}} a\\ &= a^T\frac{\partial (X^T)^{-1}}{\partial (x^T)_{ji}}X^{-1}a + a^TX^{-T}\frac{\partial X^{-1}}{\partial x_{ij}} a \end{align*}
Now plug in the formula for the derivative of $X^{-1}$.