A theorem in Linear Algebra with applications to the geometry of quadrics (Q2365713)

Let \(E\) be a finite-dimensional vector space over a field \(K\), \(f:E\to E\) a linear mapping, and \(H\) be a hyperplane in \(E\). Then a new linear mapping \(\overline f\) is obtained by restricting \(f\) to \(H\) and then projecting onto \(H\) parallel to some given direction. The author investigates the relation between the characteristic polynomials of \(f\) and \(\overline f\). As some applications for quadrics, he obtains a complete set of orthogonal invariants for quadrics in \(\mathbb{R}^ n\). In particular, two invariants which apply to quadrics of less than full rank are given.