Remark 3

One assumes that $A = K$ or $A = K(i)$ and that $E$ is finite-dimensional. Let $\Phi_1, \Phi_2$ be two positive hermitian forms on $E$, and let $V = (\alpha_{ij}), W = (\beta_{ij})$ be the matrices of these forms with respect to the same basis $(e_i)$ of $E$. Show that the hermitian form $\Phi$ whose matrix $(\gamma_{ij})$ with respect to $(e_i)$ is such that $\gamma_{ij} = \alpha_{ij}\beta_{ij}$ for every pair of indices, is positive; moreover, if $\Phi_1$ and $\Phi_2$ are non-degenerate, the same is true of $\Phi$. (In the calculation of $\Phi(x, x)$, express the $\alpha_{ij}$ in terms of the values $\Phi_1(c_i, c_i)$ for an orthogonal basis $(c_i)$ of $E$ relative to $\Phi_1$.)