Riccati-like flows and matrix approximations

A classical problem in matrix analysis, total least squares estimation and model reduction theory is that of finding a best approxiniant of a given matrix by lower rank ones. It is common believe that behind every such least squares problem there is an algebraic Riccati equation. In this paper we consider the task of minimizing the distance function fA(X) = | |A —A'|| on varieties of fixed rank symmetric matrices, using gradient-like flows for the distance function f_A. These flows turn out to have similar properties as the dynamic Riccati equation and are thus termed Riccati-like flows. A complete phase portrait analysis of these Riccati-like flows is presented, with special emphasis on positive semidefinite solutions. A variable step-size discretization of the flows is considered. The results may be viewed as a prototype for similar investigations one would like to pursue in model reduction theory of linear control systems.