Parameterization by polytopes of intersections of orbits by conjugation Original Research Article

Let S be an n×n real symmetric matrix with spectral decomposition S=QTΛQ, where Q is an orthogonal matrix and Λ is diagonal with simple spectrum {λ1,…,λn}. Also let image and image be the orbits by conjugation of S by, respectively, orthogonal matrices and upper triangular matrices with positive diagonal. Denote by image the intersection image . We show that the map image taking S′=(Q′)TΛQ′ to diag(Q′Λ(Q′)T) is a smooth bijection onto its range image , the convex hull of some subset of the n! points {(λπ(1),λπ(2),…,λπ(n)),π isapermutation}. We also find necessary and sufficient conditions for image to have n! vertices.