In this talk, which presents joint work with S. Drakunov and M. Kinyon, I will discuss the stabilization of kinematic nonholonomic systems, and its relationship with the theory of isospectral matrix flows. In particular I shall consider nonholonomic systems in the general canonical form suggested by Brockett. Nonholonomic systems are not stabilizable by smooth feedback and I will present an algorithm which uses discrete switching between smooth flows. The switching is essentially between flows which preserve eigenvalues of matrices (isospectral flows, similar to those found in the theory of integrable systems), and certain double bracket flows, similar to those found in gradient systems theory. Our double bracket flows are related to, but different from, those used by Brockett, the presenter, and others in least squares identification problems. I will discuss a generalization of these ideas to a class of systems based on the theory of Lie algebras.