Application of the companion factorization to linear non-autonomous area-preserving maps

Abstract

For a large class of discrete matrix difference equations many qualitative problems remain unsolved. The companion matrix factorization is applied here to the shift matrices associated to linear non-autonomous area-preserving maps. It permits us to introduce second order linear difference equations, which provide a faster computation of the transition matrices with respect to numerical algorithms based on the standard product of matrices. In addition, compact representations for the main elements of these discrete planar systems can be provided when using the well-known solutions of linear difference equations. Some properties and applications of current interest are presented.

Keywords:

• companion matrix factorization
• difference equations
• nested sums
• numerical algorithms
• stability
• transition matrix

AMS Subject Classifications::

• 15A23
• 37C30
• 37E99
• 39A05
• 47B39
• 65F99

Acknowledgements

The authors express their sincere thanks to the reviewers for their comments and remarks which resulted in an improvement of this article. This work was partially supported by a research grant of the UPM-CAM in Madrid, Spain.