How can you prove the following statement?
Assume $G=\left(g_{i,j}\right)$ is a random $(p\times p)$ matrix with independent elements, where $$g_{i,j}\sim N(0,1)\;\;\;\text{if }\;\;\;i<j,$$
$$g_{i,i}\sim N(0,2)\;\;\;\text{if }\;\;\;i=j,$$
$$g_{j,i}\sim N(0,1)\;\;\;\text{if }\;\;\;i<j,$$ $V$ denotes an arbitrary covariance matrix and $U=V^{\frac{1}{2}}GV^{\frac{1}{2}}.$ In this case, $$\varphi_{U}\left(A\right)=e^{-2\cdot tr\left(AVAV\right)},$$ where $$\varphi_{U}\left(A\right)=\mathbb{E}\left(e^{i\cdot tr\left(AU\right)}\right)$$ is the characteristic function of the random $U$ matrix $(A\in\mathbb{R}^{p\times p})$.