An equivalent reduction of a 2-D symmetric polynomial matrix N. P. Karampetakis Department of Mathematics Aristotle University of Thessaloniki Thessaloniki 54006, Greece URL :

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

Problem Statement – 1-D polynomial matrices

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.

Problem Statement – 1-D polynomial matrices Solution of Problem 2 Higham et.al 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 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.

Problem Statement – 2-D polynomial matrices

Zero Coprime Equivalence transformation

Invariants of ZCE

Zero Coprime System Equivalence transformation

2-D polynomial matrix reduction procedure

2-D polynomial matrix reduction procedure - Example

2-D polynomial matrix reduction procedure

2-D polynomial matrix reduction procedure - Example

2-D polynomial matrix reduction procedure

2-D polynomial matrix reduction procedure - example

2-D polynomial matrix reduction procedure

2-D polynomial matrix reduction procedure - Example

2-D polynomial matrix reduction procedure

2-D polynomial matrix reduction procedure - Example

2-D polynomial matrix reduction procedure

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

2-D polynomial matrix reduction procedure - example

2-D reduction of polynomial system matrices

Implementation in Mathematica

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 ?

Illustrative Example

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

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

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.

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

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

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

Motivation – 1-D polynomial matrices