If $A$ is an $n \times n$ invertible Hermitian matrix, let

$$U_{A}=\left\{U \in M_{n \times n}(\mathbb{C}) \mid \bar{U}^{T} A U=A\right\}$$

Show that $U_{A}$ with the operation of matrix multiplication is a group, and that det $U$ has norm 1 for any $U \in U_{A}$. What is the relation between $U_{A}$ and the complex Hermitian form defined by $A$ ?

If $A=I_{n}$ is the $n \times n$ identity matrix, show that any element of $U_{A}$ is diagonalizable.