Skew-Hamiltonian matrix

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

Choose a basis ${\displaystyle e_{1},...e_{2n}}$ in V, such that ${\displaystyle \Omega }$ is written as ${\displaystyle \sum _{i}e_{i}\wedge e_{n+i}}$. Then a linear operator is skew-Hamiltonian with respect to ${\displaystyle \Omega }$ if and only if its matrix A satisfies ${\displaystyle A^{T}J=JA}$, where J is the skew-symmetric matrix

${\displaystyle J={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}}}$

and I_n is the ${\displaystyle n\times n}$ identity matrix.^[1] 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.^[1][2]

Notes

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This page was last edited on 12 August 2023, at 18:35