4
$\begingroup$

Consider the following real symmetric matrix

$M=\left[\begin{array}{ccc} A & B\\ B^T & D \end{array}\right]$

Both A and D are real symmetric $n\times n$ matrices. B is a real $n\times n$ matrix but not necessarily symmetric. I am interested in diagonalizing the matrix with a symplectic matrix R satisfying

$R^TJR=J$

where

$J=\left[\begin{array}{ccc} 0 & 1\\ -1 &0 \end{array}\right]$

such that $R^TMR$ is block-diagonal. In particular, I am interested in an algorithm that can be coded using, say Matlab. I am aware of the following similar posts. 

1.  This post discussed a similar question for Hamiltonian matrix. However, in the above case, matrix M may not anti-commute with J. I am thinking along the line that if it is possible to similar transform M into a Hamiltonian matrix. So I guess the question boils down to how to transform a real symmetric matrix into a Hamiltonian matrix, if possible.

2. This post gives criteria that if JM is diagonalizable, then the above procedure exists. However a clear procedure is still lacking, say to conjugate JM into the Cartan subalgebra as suggested, if we assume JM is diagonalizable. Any help is appreciated.

3. I am aware of the Willianmson's theorem from This post. But in this case, M may not be positive definite.

Any help is appreciated, and thanks in advance.

 

$\endgroup$
1
$\begingroup$

I'm aware I'm answering 4 years after you asked, but if you still want to know, check out Appendix B of this https://arxiv.org/abs/0902.1502 or this paper https://arxiv.org/abs/2108.05364. Implementations are available here: https://github.com/softquanta/symplectic_decomposition.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.