On Ky Fan's result on eigenvalues and real singular values of a matrix

Ky Fan's result states that the real parts of the eigenvalues of an n×n complex matrix A is majorized by the real singular values of A. The converse was established independently by Amir-Moéz and Horn, and Mirsky. We extend the results in the context of complex semisimple Lie algebras. The real semisimple case is also discussed. The complex skew symmetric case and the symplectic case are explicitly worked out in terms of inequalities. The symplectic case and the odd dimensional skew symmetric case can be stated in terms of weak majorization. The even dimensional skew symmetric case involves Pfaffian.