Double Bracket Isospectral Flows

Abstract

Brockett (1988) has introduced a new class of isospectral flows on the set of real symmetric matrices which have remarkable properties. He considers the ordinary differential equation

$$\mathop H\limits^. \left( t \right) = \left[ {H\left( t \right),\left[ {H\left( t \right),N} \right]} \right],\;H\left( 0 \right) = {H_0}$$

((1.1))

where [A, B] = AB - BA denotes the Lie bracket for square matrices and N is an arbitrary real symmetric matrix. We term this the double bracket equation. Brockett proves that (1.1) defines an isospectral flow which, under suitable assumptions on N, diagonalizes any symmetric matrix H (t) for t → ∞. Also, he shows that the flow (1.1) can be used to solve various combinatorial optimization tasks such as linear programming problems and the sorting of lists of real numbers. Further applications to the travelling salesman problem and to digital quantization of continuous-time signals have been described, see Brockett (1989a; 1989b) and Brockett and Wong (1991). Note also the parallel efforts by Chu (1984b; 1984a), Driessel (1986), Chu and Driessel (1990) with applications to structured inverse eigenvalue problems and matrix least squares estimation.