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The Representation and Parametrization of Orthogonal Matrices
Ron Shepard1,2, Scott R Brozell1,2, Gergely Gidofalvi1,2
1Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne, Illinois 60439, United States.
The Journal of Physical Chemistry. A
|May 7, 2015
Summary
This study compares four methods for parameterizing orthogonal matrices, essential for numerical stability and computational efficiency in various applications. It details their practical use and mathematical properties for developers.
Area of Science:
- Numerical Linear Algebra
- Matrix Decompositions
- Computational Mathematics
Background:
- Orthogonal matrices are fundamental in various scientific and engineering fields.
- Efficient parameterization of orthogonal matrices is crucial for computational methods.
- Existing methods often lack comprehensive analysis of practical considerations.
Purpose of the Study:
- To compare four distinct representations of orthogonal matrices.
- To analyze their essential parameterizations for both square and rectangular cases.
- To evaluate practical aspects like numerical stability, continuity, and uniqueness.
Main Methods:
- Discussed four parametrizations: exponential, Householder reflector, Givens rotation, and rational Cayley transform.
- Considered Stiefel and Grassmann manifolds for distinct columns and span.
- Analyzed computation, extraction, and gradient transformations between matrix and parameter variables.
Main Results:
- Provided a comparative analysis of the four representations.
- Addressed computational aspects and parameter extraction for each method.
- Evaluated numerical stability, continuity, and uniqueness challenges.
Conclusions:
- Offers a comprehensive guide to orthogonal matrix parametrizations for researchers and developers.
- Highlights the trade-offs and practical considerations of each method.
- Aims to facilitate the development of new computational techniques.
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