# Convexity proof of log-determinant $\log\left|{\bf A}\left({\bf G}{\bf X}{\bf G}^H+{\bf B}\right)^{-1}+{\bf I}_N\right|$ with respect to $\bf X$

Assume that $${\bf A} \in {\mathbb C}^{N\times N}$$ is a positive semi-definite matrix, and $$\bf B \in {\mathbb C}^{N\times N}$$ is a positive definite matrix, i.e., $${\bf A} \succeq {\bf 0}$$ and $${\bf B} \succ {\bf 0}$$. $${\bf I}_N$$ stands for the $$N \times N$$ dimensional identity matrix. $${\bf X} \in {\mathbb C}^{N\times N}$$ is a matrix variabe and satisfies $${\bf X} \succeq {\bf 0}$$. I think function $$$$f({\bf X}) = \log\left|{\bf A}\left({\bf G}{\bf X}{\bf G}^H+{\bf B}\right)^{-1}+{\bf I}_N\right|$$$$ is convex with respect to matrix $$\bf X$$. For the simple scalar case, $$$$f(x) = \log\left(\frac{a}{|g|^2x + b} + 1\right),$$$$ where $$a \geq 0$$, $$b > 0$$. It can be readily verified that $$f(x)$$ is convex. I tried to prove the convexity of $$f({\bf X})$$ by following similar steps in which Boyd proved the concavity of $$\log |{\bf X}|$$ (as attached), but failed. Could anyone provide some relevant hints or references? I would be very appreciated.

By the way, we can simplify $$f({\bf X})$$ as follows \begin{align} f({\bf X}) & = \log\left|{\bf G}{\bf X}{\bf G}^H+{\bf A} + {\bf B}\right| - \log\left|{\bf G}{\bf X}{\bf G}^H + {\bf B}\right|\\ & = \log\left|{\bf G}^H\left({\bf A} + {\bf B}\right)^{-1}{\bf G}{\bf X}+ {\bf I}_N\right| + \log\left|{\bf A} + {\bf B}\right| - \log\left|{\bf G}^H{\bf B}^{-1}{\bf G}{\bf X} + {\bf I}_N\right| - \log\left|{\bf B}\right|. \end{align} So we only need to prove the convexity of $$$${\hat f}({\bf X}) = \log\left|{\bf G}^H\left({\bf A} + {\bf B}\right)^{-1}{\bf G}{\bf X}+ {\bf I}_N\right| - \log\left|{\bf G}^H{\bf B}^{-1}{\bf G}{\bf X} + {\bf I}_N\right|.$$$$

It has been proven in [1] that for given $${\bf A} \succeq {\bf 0},$$ $$h ({\bf X}) = \log \det({\bf A} {\bf X}^{-1} + {\bf I}_N)$$ is convex over the positive semidefinite cone. We then show that $$h ({\bf X})$$ is a monotone nonincreasing function, i.e., for any $${\bf X}_1 \succeq {\bf X}_2 \succeq {\bf 0}$$, $$h ({\bf X}_1) \leq h ({\bf X}_2)$$. If $${\bf X}_1 \succeq {\bf X}_2 \succeq {\bf 0}$$, we know that $${\bf A} {\bf X}_1^{-1} + {\bf I}_N \preceq {\bf A} {\bf X}_2^{-1} + {\bf I}_N$$. Due to the monotonicity of $$\log \det(\cdot)$$, we have $$h ({\bf X}_1) \leq h ({\bf X}_2)$$. Hence, $$h ({\bf X})$$ is a monotone nonincreasing function.
Let $$g({\bf X}) = {\bf G} {\bf X} {\bf G}^H + {\bf B}$$, which is an affine function, and $$f({\bf X}) = h(g({\bf X}))$$. Then, from the condition which guarantees convexity of a composite function, [2, Subsection 3.2.4], it is known that $$f({\bf X})$$ is convex over the positive semidefinite cone.
[1]. Kim, KK.K., ''Optimization and Convexity of $$\log \det({\bf A} {\bf X}^{-1} + {\bf I}_N)$$'', Int. J. Control Autom. Syst., pp. 1067–1070, 2019.