Remark 21

Suppose that the conditions of No. 3 are satisfied. Let $u$ be a hermitian endomorphism of $E$ and let $\prod_{i=1}^n (X-\lambda_i)$ be the decomposition into linear factors of its characteristic polynomial; suppose $\lambda_i \geqslant \lambda_{i+1}$ for $1 \leqslant i \leqslant n-1$.

a) Now prove that the greatest (resp. the least) of the eigenvalues $\lambda_i$ in $K$ is equal to the greatest (resp. the least) of the values of $\Phi(u(x), x)$ when $x$ ranges over the set of the $x \in E$ such that $\Phi(x, x) = 1$. (Reason directly, or apply Exercise 16 c) by reducing to the case where $A = K(i)$ ($§ 3$, Exercise 4).)

b) Let $\Psi_v$ be the restriction to a vector subspace $V$ of $E$ of the hermitian form $\Psi'$ associated with $u$, and let $u_v$ be the hermitian endomorphism of $V$ associated with $\Psi_v$. Now prove that $\lambda_k$ is the smallest of the largest eigenvalues of the $u_v$, when $V$ runs through the set of vector subspaces of $E$ of dimension $n-k+1$ (use prop. 5).