Group Theory, Linear Transformations, and Flows: (Some) Dynamical Systems on Manifolds
Abstract
OUTLINE: * Motivation (Realization process; A case study) * Basic Form (Similarity property; Decomposition property; Reversal property) * Matrix Groups and Group Actions * Tangent Space and Projection * Canonical Forms * Objective Functions and Dynamical Systems (Examples; Least squares) * New Thoughts. CONCLUSION: * Many operations used to transform matrices can be considered as matrix group actions; * The view unifies different transformations under the same framework of tracing orbits associated with corresponding group actions; * It is yet to be determined how a dynamical system should be defined over a group so as to locate the simplest form; * Continuous realization methods often enable us to tackle existence problems that are seemingly impossible to be solved by conventional discrete methods; * Group actions together with properly formulated objective functions can offer a channel to tackle various classical or new and challenging problems; * Some basic ideas and examples have been outlined in this talk; * New computational techniques for structured dynamical systems on matrix group will further extend and benefit the scope of this interesting topic.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 03, 2005
- Accession Number
- ADA433787
Entities
People
- Moody T. Chu
Organizations
- University of North Carolina at Chapel Hill