Exercise 1, § 3

Let E be a finite-dimensional vector space over $\mathbf{R}, \mathbf{C}$ or $\mathbf{H}$, $\Phi$ and $\Phi'$ two nondegenerate positive hermitian forms on E. Assume that $U(\Phi) \subset U(\Phi')$. Show that $\Phi$ and $\Phi'$ are proportional. (Make use of the fact that there exists an orthonormal basis $(e_1, \ldots, e_n)$ for $\Phi$ that is orthogonal for $\Phi'$. The mapping $u \in \mathscr{L}(E, E)$ such that $u(e_i) = e_j, u(e_j) = e_i, u(e_k) = e_k$ for $k \neq i, j$ belongs to $U(\Phi)$.)