I am just getting familiar with Hamiltonian matrices and therefore it is a little hard for me to understand their properties. My question is to help me prove or disaprove the following:
JAJ = , where
I know that J is skew-symmetric matrix, and what I was starting with is:
JAJ = A ( since = A).
by definition of Hamiltonian matrix: A is Hamiltonian if and only if = JA,
therefore
J = A,
J = A.
and I am kind of stuck at this point, I don't know what other properties I can apply , please help
You say that $A$ is Hamiltonian if $(JA)^T = JA$. Since for matrices $X$ and $Y$ we have $(XY)^T = Y^TX^T$, $A$ is Hamiltonian if and only if $A^TJ^T = JA$. Now, $J^T = -J$, so the last equation is equivalent to $-A^TJ = JA$. Moreover, $J^2 = -I$, so $J^{-1} = -J$. Thus, $A$ is Hamiltonian if and only if $A^TJ^{-1} = JA$, i.e., $A^T = JAJ$.