2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/63290Given a real, symmetric matrix S, we define the slice through S as being the connected component containing S of two orbits under conjugation: the first by the orthogonal group, and the second by the upper triangular group. We describe some classical constructions in eigenvalue computations and integrable systems which keep slices invariant -- their properties are clarified by the concept. We also parametrize the closure of a slice in terms of a convex polytope.7 pages, 2 figuresRings and Algebras58F07; 15A18; 15A23text