I am trying to understand whether the following expression is a convex function: $$ f\left (\mathbf{X} \right ) = \mathrm{tr}\left( \left ( \mathbf{\Lambda}+\alpha_0 \mathbf{\Psi}^T \left( I-e^\mathbf{{X}P} \right) \mathbf{\Psi} \right )^{-1}\right) $$ where $\mathbf{\Lambda,P}$ both are diagonal, square and positive definite matrices, $\alpha_0$ is a constant, $\Psi$ is an orthogonal matrix that $\mathbf{\Psi^{T}\Psi}=I$. In fact, I am trying to solve a minimization optimization problem and the function is my objective function. To solve it with a convex-optimization tool, I have to first check its convexity. $\mathbf{X}$ is the variable matrix we do not know in this function and it is a diagonal matrix, shown as below: $$ \mathbf{X} = \begin{bmatrix} x_1 & 0 & \cdots & 0 \\ 0 & x_2 & & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & x_n \end{bmatrix}, $$ and $$ e^\mathbf{XP} = \begin{bmatrix} e^{x_1 p_1} & 0 & \cdots & 0 \\ 0 & e^{x_2 p_2} & & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & e^{x_n p_n} \end{bmatrix}. $$
To show the convexity of the function $f$, the common way is to define a new function $g\left ( t \right )=f\left ( \mathbf{X}+t\mathbf{V} \right )$ and try to get the twice derivatives $\lim_{t \to 0}\frac{\partial^2 g\left ( t \right ) }{\partial t^2}$ greater or equal to zero. However, it is too complicated for me to solve it. It is very appreciate that someone can help me and thank you in advance.