I have a positive definite matrix $X$ of order $m$. We know that $det(X)=\prod_{i=1}^m \lambda_i$ and $tr(X)=\sum_{i=1}^m \lambda_i$, where $\lambda_1,\ldots,\lambda_m$ are the $m$ eigenvalues of $X$.
I need to work out the following expression $\log\{det(X)\}=\sum_{i=1}^m\log(\lambda_i)$. Playing a little bit I "found out" that if I have a function $g$ and I apply it componentwise to the elements of $X$, for brevity I will denote the trasformed matrix by $X_g$, then $tr(X_g)=\sum_{i=1}^m g(\lambda_i)$.
I have the following questions:
1) Is the last relation true? At least as long as it is possible to apply the function $g$ componentwise to the elements of $X$?
2) The first question is just for curiosity because, unfortunately, my matrix $X$ may have negative entries, so I cannot apply the logarithm componentwise. Thus, I was wondering whether there exist some viable/nice way to obtain an expression for $\log\{det(X)\}$.
Cheers
No, the desired relation does not hold. Consider $g(x) = e^x$, and take $$ X = \pmatrix{2&1\\1&2} $$ Then $tr(X_g) = 2e^2 \neq e^3 + e^1$.
If $g$ is of the form $g(x) = ax+b$, then your result does hold.