A LIE THEORETIC APPROACH TO THOMPSON'S THEOREMS ON SINGULAR VALUES–DIAGONAL ELEMENTS AND SOME RELATED RESULTS

Abstract

Thompson's famous theorems on singular values–diagonal elements of the orbit of an n×n matrix A under the action (1) U(n) [otimes ] U(n) where A is complex, (2) SO(n) [otimes ] SO(n), where A is real, (3) O(n) [otimes ] O(n) where A is real are fully examined. Coupled with Kostant's result, the real semi-simple Lie algebra [sfr ][ofr ]_{n, n} yields (2) and hence (3) and the sufficient part (the hard part) of (1). In other words, the curious subtracted term(s) are well explained. Although the diagonal elements corresponding to (1) do not form a convex set in [Copf ]ⁿ, the projection of the diagonal elements into ℝⁿ (or iℝⁿ) is convex and the characterization of the projection is related to weak majorization. An elementary proof is given for this hidden convexity result. Equivalent statements in terms of the Hadamard product are also given. The real simple Lie algebra [sfr ][ufr ]_{n, n} shows that such a convexity result fits into the framework of Kostant's result. Convexity properties and torus relations are studied. Thompson's results on the convex hull of matrices (complex or real) with prescribed singular values, as well as Hermitian matrices (real symmetric matrices) with prescribed eigenvalues, are generalized in the context of Lie theory. Also considered are the real simple Lie algebras [sfr ][ofr ]_{p, q} and [sfr ][ofr ]_{p, q}, p < q, which yield the rectangular cases. It is proved that the real part and the imaginary part of the diagonal elements of complex symmetric matrices with prescribed singular values are identical to a convex set in ℝⁿ and the characterization is related to weak majorization. The convex hull of complex symmetric matrices and the convex hull of complex skew symmetric matrices with prescribed singular values are given. Some questions are asked.