Exercise 17, § 2

a) Let $n\in \mathbf{N}$. Let E be a Hilbert space of dimension $n$ and let $u\in \mathscr{L}(E)$. Let $x\in E$ be of norm 1 such that $u(x) = 0$. For every vector $y\in E$ of norm 1, denote by $\Delta_u(y)$ the discriminant of the quadratic form defined on $y^{\circ}$ by $z\mapsto \langle z|u(z)\rangle$.

For every $y\in$ E of norm 1, prove that $|\langle x|y\rangle |^2\Delta_u(x) = \Delta_u(y)$. (Note that $\Delta_u(y) =\langle a|\bigwedge u^{n-1}(a)\rangle /\langle a|a\rangle$ for every nonzero element $a$ of the space $\bigwedge^{n-1}y^{\circ}\subset \bigwedge^{n-1}E$, the space $\bigwedge^{n-1}E$ being endowed with its canonical Hilbert structure, cf. EVT, V, p. 33; then verify the formula separately when $x$ is orthogonal to $y$ and when $x=y.$)

b) Let A be a matrix of type $(n, n)$ with complex coefficients such that $^t\overline{A}=$ A. For every integer $i$ such that $1\leqslant i\leqslant n$, let $A_i$ be the matrix of type $(n-1, n-1)$ obtained by deleting the $i^e$ row and the $i^e$ column of A. Let $(e_1, . . . , e_n)$ be an orthonormal basis of $\mathbf{C}^n$, endowed with its canonical Hilbert structure, consisting of eigenvectors for A, and let $\lambda_i$ be the eigenvalue of $u$ corresponding to $e_i$. Write $e_i= (e_{i,j})_{1\leqslant j\leqslant n}$.

For all $(i, j)$, prove that

$|e_{i,j}|^2\prod_{k\not=i}(\lambda_i-\lambda_k) =$ det($\lambda_i-A_j$).

(Reduce to the case where $\lambda_i$ = 0, then apply the preceding question to the space $\mathbf{C}^n$, to the endomorphism $u:y\mapsto Ay$ and to the vector $x=e_i.$)