Suppose $f(\alpha) = \log \det[({\bf A} - \alpha {\bf B})({\bf A} - \alpha {\bf B})^\top] + \alpha c$, where ${\bf A}$ is an $m \times n$ matrix, where $m < n$. I want to find the value of $\alpha$ where the gradient of $f$ is zero. I can use general purpose optimization routine implemented in a software to find the value of $\alpha$. However, I am thinking whether there is a closed form solution to this equation?
Define ${\bf X} = ({\bf A} - \alpha {\bf B})({\bf A} - \alpha {\bf B})^\top$ $$ f'(\alpha) = 2 \ \text{trace}({\bf X}^{-1}(\alpha {\bf BB}^\top - {\bf B A}^\top)) + c = 0 $$ I am not sure how to extract $\alpha$ from the inverse term.