Exercise 12, § 1

a) Let $\sum_{i,j} c_{ij} x_i \bar{x}_j$ where $c_{ji} = \bar{c}_{ij}$ be a nondegenerate positive hermitian form, and $\Delta$ its determinant; show that $\Delta \leq c_{11} c_{22} \ldots c_{nn}$ (express $\Delta$ and the $c_{ii}$ in terms of the eigenvalues of the hermitian form, and use prop. 2).

b) Deduce that if $(a_{ij})$ is a square matrix of order $n$ with arbitrary complex elements and $\Delta$ is its determinant, one has ("Hadamard's inequality")

with equality only if one of the terms on the right-hand side is zero or if, for all distinct indices $h, k$ (multiply the matrix $(a_{ij})$ by the conjugate of its transpose).