Abstract: The Schur-Horn convexity theorems relates the eigenvalues of a positive definite Hermitian matrix to the diagonal entries in its Gauss decomposition. This result was extended by Kostant to the context of semi-simple Lie groups and was further generalized by Atiyah-Guillemin-Sternberg in their celebrated convexity theorem. We will explain an analogous result for certain complex symmetric matrices. In addition we will discuss convexity features of a complexified polar decomposition of an invertible matrix. The work presented is partly joined with S.Gindikin and Michael Otto. On the technical side we slightly improve and extends results of Kostant, Heckmann, Atiyah, Guillemin-Sternberg, Duistermaat and Harish-Chandra.