Let $X$ be an $n \times n$ symmetric matrix, whose entries are denoted as $X_{i j}, 1 \leq i, j \leq n .$ Suppose that all the entries on and above the diagonal are independent, i.e., the entries $X_{i j}, 1 \leq i \leq j \leq n$ are independent. Further assume that $X_{i i}$ has $N(0,2)$ distribution for every $1 \leq i \leq n,$ and $X_{i j}$ has $N(0,1)$ distribution for every $1 \leq i<j \leq n .$ Now let $U$ be an $n \times n$ orthogonal matrix, and let $Y=U^{T} X U .$ Clearly $Y$ is symmetric. Prove that all the entries on and above the diagonal of $Y$ are independent, and find the distributions of all these entries.
I have no ideas and can anyone help me?