1. An equivalent reduction of a 2-D symmetric polynomial matrix N. P. Karampetakis Department of Mathematics Aristotle University of Thessaloniki Thessaloniki 54006, Greece Email : karampet@math.auth.grkarampet@math.auth.gr URL : http://anadrasis.math.auth.grhttp://anadrasis.math.auth.gr

2. Contents Preliminaries  Problem statement for 1-D polynomial matrices Finite and infinite elementary divisor structure of 1-D polynomial matrices Problem statement and solution  Problem statement for 2-D polynomial matrices Invariant polynomials & zeros of 2-D polynomial matrices Zero coprime equivalence transformation and its invariants Zero coprime system equivalence and its invariants Problem statement 2-D symmetric polynomial matrix reduction procedure 2-D polynomial system matrix reduction procedure Implementation in Mathematica Conclusions

3. Problem Statement – 1-D polynomial matrices

7. Motivation Numerical methods that ignore the special structure of the polynomial matrix T(s) (like the companion form) often destroy these qualitatively important spectral symmetries, sometimes even to the point of producing physically meaningless or uninterpretable results. Storage and computational cost are reduced if a method that exploits symmetry is applied. i.e. The solution of Ax=B, with A symmetric, via a symmetric banded solver uses O(n) storage and O(n) flops, while using LU methods that not exploits symmetry uses O(n 2 ) storage and O(n 3 ) flops.

8. Problem Statement – 1-D polynomial matrices Solution of Problem 2 Higham et.al. 2005 Symmetric linearizations for matrix polynomials.  The reduction is used for the solution of the polynomial eigenvalue problem T(s)x=0.  A vector space of symmetric pencils sE-A is generated with eigenvectors closely related to those of T(s).  No transformation is used. The matrix pencil proposed by Antoniou and Vologiannidis is not in the vector space of symmetric pencils proposed by Higham et.al.. Antoniou and Vologiannidis 2004. Linearizations of polynomial matrices with symmetries and their applications.  One specific symmetric linearization is proposed.  The new matrix pencil sE-A is connected with T(s) through a unimodular equivalence relation.

9. Problem Statement – 2-D polynomial matrices

12. Zero Coprime Equivalence transformation

13. Invariants of ZCE

14. Zero Coprime System Equivalence transformation

15. 2-D polynomial matrix reduction procedure

16. 2-D polynomial matrix reduction procedure - Example

17. 2-D polynomial matrix reduction procedure

19. 2-D polynomial matrix reduction procedure - Example

20. 2-D polynomial matrix reduction procedure

21. 2-D polynomial matrix reduction procedure - example

24. 2-D polynomial matrix reduction procedure

25. 2-D polynomial matrix reduction procedure - Example

26. 2-D polynomial matrix reduction procedure

27. 2-D polynomial matrix reduction procedure - Example

29. 2-D polynomial matrix reduction procedure

30. 2-D polynomial matrix reduction procedure - example Define the following companion form

31. 2-D polynomial matrix reduction procedure - example

32. 2-D reduction of polynomial system matrices

38. Implementation in Mathematica

41. Conclusions  A two-stage algorithm, easily implementable in a computer symbolic environment, has been provided for the reduction of a 2-D symmetric polynomial matrix to a zero coprime equivalent 2-D symmetric matrix pencil.  The results has also been adapted to 2-D system matrices.  Advantage.  We can use existing robust numerical algorithms for 2-D matrix pencils in order to compute structural invariants of 2-D symmetric polynomial matrices  Disadvantage.  The size of the matrices that we use.  An implementation of this algorithm in the package MATHEMATICA accompanied with one example is given.  Further research  Reduction of symmetric and positive definite polynomial matrices.  Use of other matrix pencil reduction methods (Higham et.al.).  New numerical techniques for investigating structural invariants of 2-D symmetric matrix pencils.  Infinite elementary divisor structure ?

42. Illustrative Example

46. Motivation – 1-D polynomial matrices we get Consider the homogeneous system Then by defining the following variables or equivalently

47. Motivation – 1-D polynomial matrices is known as the first companion form of T(ρ). and the following matrix pencil

48. Motivation – 1-D polynomial matrices in the sense that the compound matrices Note that the following extended unimodular equivalent transformation connects the polynomial matrix T(ρ) and the respective pencil ρE-A. do not lose rank in C. Conclusion. Since, T(ρ) and ρE-A are e.u.e. they possess the same finite elementary divisor structure.

49. Motivation – 1-D symmetric polynomial matrices Disandvantages. Numerical methods that ignore the special structure of the polynomial matrix T(ρ) (like the ones above) often destroy these qualitatively important spectral symmetries, sometimes even to the point of producing physically meaningless or uninterpretable results. Storage and computational cost are reduced if a method that exploits symmetry is applied i.e. The solution of Ax=B, with A symmetric, via a symmetric banded solver uses O(n) storage and O(n) flops, while using LU methods that not exploits symmetry uses O(n 2 ) storage and O(n 3 ) flops. Consider the symmetric polynomial matrix and the respective e.u.e. matrix pencil

50. Motivation – 1-D symmetric polynomial matrices we get the following homogeneous system Consider the homogeneous system Then by defining the following variables

51. Motivation – 1-D polynomial matrices is e.u.e. to ρE-A. where the symmetric matrix pencil

52. Motivation – 1-D polynomial matrices