September 2008CORTONA-ITALY, 20081 DOUBLY STRUCTURED SETS OF SYMPLECTIC MATRICES Froilán M. Dopico Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM.

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September 2008CORTONA-ITALY, DOUBLY STRUCTURED SETS OF SYMPLECTIC MATRICES Froilán M. Dopico Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM and Departamento de Matemáticas, Universidad Carlos III de Madrid, Spain Joint work with Charles R. Johnson, The College of William and Mary, Williamsburg, VA, USA

September 2008CORTONA-ITALY, Symplectic Matrices

September 2008 CORTONA-ITALY, The problem to be solved Symplectic matrices are implicitly defined as solutions of the nonlinear matrix equation This characterization makes difficult to work with them, both in theory and in structured numerical algorithms. OUR GOAL: To present an explicit description (parametrization) of the group of symplectic matrices, i.e., to find the set of solutions of and to apply this parametrization to construct symplectic matrices that have extra structures. This is useful for checking if a matrix is symplectic, but not for constructing symplectic matrices.

September 2008CORTONA-ITALY, Outline of the talk 1.Previous results 2.Parametrization 3.Brief Summary on subparametrization problems 4.Description of doubly structured sets (symplectic and other property): LU factorizations of symplectic Orthogonal symplectic Positive definite symplectic Positive elementwise symplectic TN, TP, oscillatory symplectic Symplectic M-Matrices 5. Conclusions and Open problems

September 2008CORTONA-ITALY, Previous I: A result by Mehrmann (SIMAX, 1988)

September 2008CORTONA-ITALY, Parametrization with nonsingular (1,1)-block

September 2008CORTONA-ITALY, Parametrization with nonsingular (1,1)-block

September 2008CORTONA-ITALY, Previous II: The complementary bases theorem

September 2008CORTONA-ITALY, The group of symplectic matrices

September 2008CORTONA-ITALY, Subparametrization Problems (I)

September 2008CORTONA-ITALY, Subparametrization Problems (II)

September 2008CORTONA-ITALY, LU factorizations of Symplectic Matrices (I)

September 2008CORTONA-ITALY, LU factorizations of Symplectic Matrices (II)

September 2008CORTONA-ITALY, Symplectic Orthogonal Matrices

September 2008CORTONA-ITALY, Symplectic Positive Definite (PD) Matrices

September 2008CORTONA-ITALY, Symplectic Matrices with positive entries

September 2008CORTONA-ITALY, Totally Nonnegative (TN) Symplectic Matrices (I) DEFINITIONS:

September 2008CORTONA-ITALY, Totally Nonnegative (TN) Symplectic Matrices (II)

September 2008CORTONA-ITALY, Totally Nonnegative (TN) Symplectic Matrices (III)

September 2008CORTONA-ITALY, Totally Nonnegative (TN) Symplectic Matrices (IV)

September 2008CORTONA-ITALY, Symplectic M-Matrices (I)

September 2008CORTONA-ITALY, Symplectic M-Matrices (II)

September 2008CORTONA-ITALY, Conclusions and Open Problems An explicit description of the group of symplectic matrices has been introduced. It allows us to characterize very easily several structured sets of symplectic matrices. How to extend this approach to other structured sets? Rank structured symplectic matrices? Perturbation theory with respect the symplectic parameters? Interesting properties? How to compute the parametrization in a stable an efficient way if we are given the entries of a symplectic matrix? Have these symplectic parameters an intrinsic meaning?