Unduloid

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 $\mathrm{\Omega }$. Such a space must be even-dimensional. A linear map $A:V\mapsto V$ is called a skew-Hamiltonian operator with respect to $\mathrm{\Omega }$ if the form $x,y\mapsto \mathrm{\Omega }(A(x),y)$ is skew-symmetric.

Choose a basis $e_1,...e_{2n}$ in V, such that $\mathrm{\Omega }$ is written as $\sum _i{e}_i\wedge e_{n+i}$. Then a linear operator is skew-Hamiltonian with respect to $\mathrm{\Omega }$ if and only if its matrix A satisfies $A^{T}J=JA$, where J is the skew-symmetric matrix

$J=\left[\begin{array}{cc}0& I_n\\ -I_n& 0\\ \end{array}\right]$

and I_n is the $n\times n$ identity matrix. Such matrices are called skew-Hamiltonian.

The square of a Hamiltonian matrix is skew-Hamiltonian. The converse is also true: every skew-Hamiltonian matrix can be obtained as the square of a Hamiltonian matrix.

Notes

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