A differential equation for diagonalizing complex semisimple Lie algebra elements

In this paper, we consider a generalization of Ebenbauer’s differential equation for non-symmetric matrix diagonalization to a flow on arbitrary complex semisimple Lie algebras. The flow is designed in such a way that the desired diagonalizations are precisely the equilibrium points in a given Cartan subalgebra. We characterize the set of all equilibria and establish a Morse–Bott type property of the flow. Global convergence to single equilibrium points is shown, starting from any semisimple Lie algebra element. For strongly regular initial conditions, we prove that the flow converges to an element of the Cartan subalgebra and thus achieves asymptotic diagonalization.