I'm new here, so "Hi" to everyone :D
I got the following problem. I have the matrices $A$, $B$, $C$, $X$ and $Y$. All matrices are square (say n-by-n). In particular: - $A$ is full rank - $B$ is symmetric and (semi)definite positive; - $C$ is diagonal and definite positive; - $Y$ is diagonal and definite positive; - $X$ is diagonal ($X = \operatorname{diag}\{x_1, \ldots,x_n\}$) and it is the unknown matrix;
Then I have the following function: $f(X) = (A(B+X^{T}YX)^{-1}A^{T} + C)^{-1}$ (it may seem dumb to write $X^{T}$ since it is diagonal, but I think this is the best way to write it).
I would like to evaluate the derivative of the trace of $f(X)$ with respect to each $x_i$.
Any idea?