论文信息 - Complementary bases in symplectic matrices and a proof that their determinant is one

Complementary bases in symplectic matrices and a proof that their determinant is one

New results on the patterns of linearly independent rows and columns among the blocks of a symplectic matrix are presented. These results are combined with the block structure of the inverse of a symplectic matrix, together with some properties of Schur complements, to give a new and elementary proof that the determinant of any symplectic matrix is +1. The new proof is valid for any field. Information on the zero patterns compatible with the symplectic structure is also presented.

Charles R. Johnson | Froilán M. Dopico | F. M. Dopico | F. Dopico

[1] F. Tisseur,et al. STRUCTURED TOOLS FOR STRUCTURED MATRICES , 2003 .

[2] V. Mehrmann. The Autonomous Linear Quadratic Control Problem: Theory and Numerical Solution , 1991 .

[3] H. Faßbender. Symplectic Methods for the Symplectic Eigenproblem , 2002, Springer US.

[4] Charles R. Johnson,et al. Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[5] Volker Mehrmann,et al. Canonical forms for Hamiltonian and symplectic matrices and pencils , 1999 .

[6] Claude Viterbo,et al. An introduction to symplectic topology , 1991 .

[7] Niloufer Mackey,et al. On the Determinant of Symplectic Matrices , 2003 .

[8] P J Fox,et al. THE FOUNDATIONS OF MECHANICS. , 1918, Science.

[9] Volker Mehrmann,et al. A Chart of Numerical Methods for Structured Eigenvalue Problems , 1992, SIAM J. Matrix Anal. Appl..

[10] Jerrold E. Marsden,et al. Foundations of Mechanics, Second Edition , 1987 .

[11] V. Arnold. Mathematical Methods of Classical Mechanics , 1974 .