Advanced Studies in Pure Mathematics

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

Yoshimasa Nakamura

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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

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

Received: 11 March 1991
First available in Project Euclid: 15 August 2018

Permanent link to this document euclid.aspm/1534359527

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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.

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