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In linear algebra, skew-Hamiltonian matrices are special matrices which correspond to skew-symmetric bilinear forms on a symplectic vector space.
Let V be a vector space, equipped with a symplectic form Ω {\displaystyle \Omega }. Such a space must be even-dimensional. A linear map A : V ↦ V {\displaystyle A:\;V\mapsto V} is called a skew-Hamiltonian operator with respect to Ω {\displaystyle \Omega } if the form x , y ↦ Ω , y ] {\displaystyle x,y\mapsto \Omega ,y]} is skew-symmetric.
Choose a basis e 1 , . . . e 2 n {\displaystyle e_{1},...e_{2n}} in V, such that Ω {\displaystyle \Omega } is written as ∑ i e i ∧ e n + i {\displaystyle \sum _{i}e_{i}\wedge e_{n+i}}. Then a linear operator is skew-Hamiltonian with respect to Ω {\displaystyle \Omega } if and only if its matrix A satisfies A T J = J A {\displaystyle A^{T}J=JA} , where J is the skew-symmetric matrix
and In is the n × n {\displaystyle n\times n} identity matrix. Such matrices are called skew-Hamiltonian.