symplectic matrix

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symplectic matrix

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symplectic matrix (plural symplectic matrices)

1. (linear algebra) For given field F (especially the real numbers), even order 2n and nonsingular skew-symmetric matrix Ω, any 2n×2n matrix M with elements in F such that M^TΩM = Ω (where M^T denotes the transpose of M).
   • 2001, Alberto Abbondandolo, Morse Theory for Hamiltonian Systems‎, CRC Press (Chapman & Hall/CRC), page x:

     Then we define the Krein signature of the eigenvalues of modulus one of a symplectic matrix and use this concept to build the rotation function on the symplectic group.

   • 2002, Heike Fassbender, Symplectic Methods for the Symplectic Eigenproblem, Kluwer Academic, page 173:

     In the previous chapter algorithms for computing the eigenvalues of symplectic matrices have been considered that are based on an elimination process for computing the butterfly form of symplectic matrix. Unfortunately, this approach is not suitable when dealing with large and sparse symplectic matrices as an elimination process can not make full use of the sparsity.

   • 2012, Dario A. Bini, Bruno Iannazzo, Beatrice Meini, Numerical Solution of Algebraic Riccati Equations, Society for Industrial and Applied Mathematics, page 121:

     In order to fully exploit these structures we rely on the properties of Hamiltonian, skew-Hamiltonian, and symplectic matrices discussed in Section 1.5. In Section 4.1 we introduce the elementary symplectic matrices which are used by the structured algorithms.

Ω is often chosen to be the block matrix ${\displaystyle \textstyle \Omega ={\begin{bmatrix}0&I_{n}\\-I_{n}&0\end{bmatrix}}}$, where I_n is the n×n identity matrix.