Parametrization by polytopes of intersections of orbits by conjugation
نویسنده
چکیده
Let S be an n×n real symmetric matrix with spectral decomposition S = Q ΛQ, where Q is an orthogonal matrix and Λ is diagonal with simple spectrum {λ1, . . . , λn}. Also let OS e RS be the orbits by conjugation of S by, respectively, orthogonal matrices and upper triangular matrices with positive diagonal. Denote by FS the intersection OS ∩RS . We show that the map F : F̄S → R n taking S′ = (Q) ΛQ′ to diag (Q′ Λ (Q) ) is a smooth bijection onto its range PS, the convex hull of some subset of the n! points {(λπ(1), λπ(2), . . . , λπ(n)), π is a permutation}. We also find necessary and sufficient conditions for PS to have n! vertices.
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