Lie series and Riccati equations

Abstract

Lie series and a special matrix notation for first-order differential operators are used to show that the Lie group properties of matrix Riccati equations arise in a natural way. The Lie series notation makes it evident that the solutions of the matrix Riccati equations are curves in a group of nonlinear transformations which is a generalization of the linear fractional transformations familiar from classical complex analysis. It is easy to obtain a linear representation of the Lie algebra of the group of nonlinear transformations, and this linearization leads directly to the standard linearization of the matrix Riccati equations. We note that the matrix Riccati equations considered here are of the general rectangular type.

This work has been partially supported by the Consejo Nacional de Ciencia y Tecnología (CONACyT, México) Project ICCBCHE 790373 (covering the stay of L. Hlavatý at IIMAS), and partially by the National Science Foundation (NSF, USA) grant # MCS-8102083 (covering the sabbatical leave of S. Steinberg).

On leave from the Institute of Physics, Czechoslovak Academy of Sciences, Prague, ČSSR.

Presentor.

On leave from the Department of Mathematics and Statistics, The University of New Mexico, Albuquerque N. M., USA.

This work appeared as a preprint of the University of New Mexico, and will be published in Journal of Mathematical Analysis and Applications.