Given a matrix $A\in M_{k\times l}(\mathbb{C})$ we define the hermitian transpose of $A$ as the matrix $A^*=\overline{A}^t\in M_{l\times k}(\mathbb{C})$. We say a matrix $H\in M_k(\mathbb{C})$ is hermitian if $H=H^*$. In particular, we can show easily that all eigenvalues of an hermitian matrix are real. We say that a $3\times 3$ hermitian matrix is of signature (2,1) if it has two positive eigenvalues and one negative eigenvalue. Given two hermitian matrices $H$ and $H'$, if $C$ is a matrix such that $$H'=C^*HC,$$ we say $C$ is a Cayley transform.
I'm reading the Notes on Complex Hyperbolic Geometry by John Parker, and at the page 8 of it he says (at least it seems so) that if $H$ and $H'$ are hermitian matrices of signature (2,1) then such a $C$ always exists. Is that so? How can one show it?
I've tried to solve it by system of equations, but it's not so simple... :(
This is a special case of Sylvester's law of inertia. By the way, the term Cayley transform usually refers to something else.