Suppose I have the following maximization problem:
$\log\det(\alpha K_p)-c\alpha$ with respect to $\alpha$ with $c$ being a constant and $m$ being the dimension of $K_p$. Here, one of the eigenvalues of $K_p$ is $0$ and so $\det(\alpha K_p)=0.$
Am I allowed to do the following
$\log\det(\alpha K_p)-c\alpha = \log(\alpha^m\det(K_p)) - c\alpha = \log \alpha^m + \log\det(K_p) - c\alpha$
and disregard $\log(\det(K_p))$ as it is a constant (even though it's value is infinity) ?
My purpose of doing this is to that there is a closed form expression to the optimal value of $\alpha$...