Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems

AbstractWe establish a number of properties associated with the dynamical system Ḣ = [H,[H, N]], where H and N are symmetric n by n matrices and [A, B] = AB − BA. The most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices. We are especially interested in the role of this equation as an analog computer. For example, we show how to map the data associated with a linear programming problem into H(0) and N in such a way as to have Ḣ = [H[H, N]] evolve to a solution of the linear programming problem. This result can be applied to find systems which solve a variety of genetic combinatorial optimization problems, and it even provides an algorithm for diagonalizing symmetric matrices