Gradient Flows, Convexity, and Adjoint Orbits

Abstract

This dissertation studies some matrix results and gives their Generalizations in the context of semisimple Lie groups. The adjoint orbit is the primary object in our study. The dissertation consists of four chapters. Chapter 1 is a brief introduction about the interplay between matrix theory and Lie theory. In Chapter 2 we introduce some structure theory of semisimple Lie groups and Lie algebras. It involves the root space decompositions for complex and real semisimple Lie algebras, Cartan decomposition and Iwasawa decomposition for real semisimple Lie algebras and Lie groups. They play significant roles in our generalizations. In Chapter 3 we introduce a famous problem on Hermitian matrices proposed by H. Weyl in 1912, which has been completely solved. Motivated by a recent paper of Li et al. [34] we consider a generalized problem in the context of semisimple as well as reductive Lie groups. We give the gradient flow of a function corresponding to the generalized problem. This provides a unified approach to deriving several results in [34]. Chapter 4 is essentially a brief survey on some generalized numerical ranges associated with Lie algebras. The classical numerical range of an n×n complex matrix is the image of the unit sphere in Cnunder the quadratic form. One of the most beautiful properties is that the numerical range of a matrix is always convex. It is known as the Toeplitz-Hausdorff theorem. We give another proof of the convexity of some generalized numerical range associated with a compact Lie group. The Toeplitz-Hausdorff theorem becomes a special case.