Let

$g(boldsymbol{theta},boldsymbol{theta_0}) = trace (

boldsymbol{Omega{(boldsymbol{theta})}}^{-1} boldsymbol{Omega{(boldsymbol{theta_0})}})-ln(det(boldsymbol{Omega{(boldsymbol{theta_0})}})/det(boldsymbol{Omega{(boldsymbol{theta})}}))-N $

where $boldsymbol{theta} in boldsymbol{Theta}$ with $boldsymbol{Theta}$ a compact subset of $R^{n}$, $n$ and $N$ are fixed numbers, and $boldsymbol{theta_0}$ belongs to the interior of $boldsymbol{Theta}$.

Denote the eigenvalues of the symmetric matrices $boldsymbol{Omega{(boldsymbol{theta_0})}}$ and $boldsymbol{Omega{(boldsymbol{theta})}}$ by $lambda_{0s}$ and $lambda_s$ $(s=1,2,…,N)$ respectively, where $lambda_{0s}>0$ and $lambda_s>0$ for all $s$.

Given the above, does the following hold or is a further condition required, and if so which one?

$g(boldsymbol{theta},boldsymbol{theta_0}) = sum_{s=1}^N ((lambda_{0s}/lambda_{s})-ln(lambda_{0s}/lambda_{s})-1) ?$