The sums of symplectic, Hamiltonian, and skew-Hamiltonian matrices
| Autor: | Ralph John de la Cruz, Agnes T. Paras |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Numerical Analysis
Algebra and Number Theory 010102 general mathematics 010103 numerical & computational mathematics 01 natural sciences Combinatorics Matrix (mathematics) symbols.namesake symbols Discrete Mathematics and Combinatorics Geometry and Topology 0101 mathematics Hamiltonian (quantum mechanics) Mathematics::Symplectic Geometry Eigenvalues and eigenvectors Symplectic geometry Mathematics |
| Zdroj: | Linear Algebra and its Applications. 603:84-90 |
| ISSN: | 0024-3795 |
| DOI: | 10.1016/j.laa.2020.05.036 |
| Popis: | A complex 2 n × 2 n matrix A is called skew-Hamiltonian, Hamiltonian, and symplectic if A J = A , A J = − A , and A J = A − 1 , respectively, in which J = [ 0 I n − I n 0 ] and A J = J − 1 A T J . We prove that each 2 n × 2 n matrix is a sum of type “symplectic + Hamiltonian”. A 2 n × 2 n matrix A is a sum of type “symplectic + symplectic” if and only if A A J is similar to A J A . A 2 n × 2 n matrix A is a sum of type “symplectic + skew-Hamiltonian” if and only if the Jordan blocks of A − A J with eigenvalue 2i and size k≥ 2 come in pairs of the form J k ( 2 i ) ⊕ J k ( 2 i ) and J k ( 2 i ) ⊕ J k + 1 ( 2 i ) . |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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