## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Progress in Differential Geometry, K. Shiohama, ed. (Tokyo: Mathematical Society of Japan, 1993), 213 - 229

### Lax Equations Associated with a Least Squares Problem and Compact Lie Algebras

#### Abstract

The gradient flow in a least squares problem on a Lie group takes a Lax form [8]. We associate the Lax equation with homogeneous spaces and symmetric spaces of compact simple Lie groups. The critical points of the Lax equation lie in the Cartan subalgebras of the simple Lie algebras. A reduction from homogeneous spaces to symmetric spaces is described by a ‘coalescence’ of roots. For the complex Grassmann manifold, it is shown that an initial value problem of the Lax equation can be uniquely solved. Some applications to a least squares fitting problem and a linear programming problem are discussed.

#### Article information

**Source***Progress in Differential Geometry*, K. Shiohama, ed. (Tokyo: Mathematical Society of Japan, 1993), 213-229

**Dates**

Received: 11 March 1991

First available in Project Euclid:
15 August 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1534359527

**Digital Object Identifier**

doi:10.2969/aspm/02210213

**Mathematical Reviews number (MathSciNet)**

MR1274950

**Zentralblatt MATH identifier**

0799.58031

#### Citation

Nakamura, Yoshimasa. Lax Equations Associated with a Least Squares Problem and Compact Lie Algebras. Progress in Differential Geometry, 213--229, Mathematical Society of Japan, Tokyo, Japan, 1993. doi:10.2969/aspm/02210213. https://projecteuclid.org/euclid.aspm/1534359527