How to diagonalize a Hermitian matrix using a quasi-unitary matrix?

Question

I met a problem requiring the diagonalization of a $2n\times 2n$ Hermitian matrix $H$ in the following way:

$U^{*} HU=D$,

where $D$ is diagonal, $U^*$ is the transpose conjugate of $U$. The matrix $U$ is restricted by

$UJU^*=J$,

here $J=diag(I_n,-I_n)$ with $I_n$ is the $n\times n$ identity matrix. Namely, $U$ is quasi-unitary.

I know how to find $D$, by diagonalizing $HJ$ using a similarity transformation, one will find

$V^{-1}HJV=JD$.

But I cannot obtain $U$ from the above $V$. How can we find the matrix $U$?

Answer 1

Why can't you multiply by $J^{-1}$? Is JV not quasi-unitary?

What is the matrix V? What properties does it have? How did you get to this factorization?