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DOUBLY STRUCTURED SETS OF SYMPLECTIC MATRICES

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.

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DOUBLY STRUCTURED SETS OF SYMPLECTIC MATRICES

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  1. 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 CORTONA-ITALY, 2008

  2. Symplectic Matrices CORTONA-ITALY, 2008

  3. The problem to be solved Symplectic matrices are implicitly defined as solutions of the nonlinear matrix equation This is useful for checking if a matrix is symplectic, but not for constructing symplectic matrices. 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. CORTONA-ITALY, 2008

  4. Outline of the talk • Previous results • Parametrization • Brief Summary on subparametrization problems • 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 CORTONA-ITALY, 2008

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

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

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

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

  9. The group of symplectic matrices CORTONA-ITALY, 2008

  10. Subparametrization Problems (I) CORTONA-ITALY, 2008

  11. Subparametrization Problems (II) CORTONA-ITALY, 2008

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

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

  14. Symplectic Orthogonal Matrices CORTONA-ITALY, 2008

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

  16. Symplectic Matrices with positive entries CORTONA-ITALY, 2008

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

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

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

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

  21. Symplectic M-Matrices (I) CORTONA-ITALY, 2008

  22. Symplectic M-Matrices (II) CORTONA-ITALY, 2008

  23. 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? CORTONA-ITALY, 2008

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