Form for pseudo-unitary matrices of particular dimension

Question

I know that the unimodular pseudo-unitary group is a Lie group defined by $$\text{SU}(p, q)= \{M \in \text{SL}_{p+q}(\mathbb{C}): MAM^{*} = A \} \text{,}$$ where $A= \begin{pmatrix} 1_p & 0 \\ 0 & -1_{q} \\ \end{pmatrix}$, $M$ is a complex $(p+q) \times (p+q)$ matrix of determinant $1$ with $p$ the number of $1$s and $q$ the number of $-1$s in the diagonal entries of $A$, and $M^{*}$ denotes the conjugate transpose of $M$. Moreover, when $p=q=1$, then $M$ has the following form: $$M= \begin{pmatrix} \alpha & \beta \\ \bar{\beta} & \bar{\alpha} \end{pmatrix}, \alpha, \beta \in \mathbb{C}, |\alpha|^2 - |\beta|^2=1 \text{,}$$ where $\bar{}$ denotes complex conjugation. My question is what would the form of $M$ be if $p=1,q=2$ or $p=1,q=3$? What about for general $p,q$? I haven't found any useful references in that regard.

Answer 1

There is a partial parametrization of the set of pseudo-unitary matrices $U(p,q)=\{M;M^*AM=A\}$. Def. We say that $K$ is pseudo-hermitian ($K\in PH$) iff $K^*A=AK$.

Note that the matrices $K$ are solutions of a linear system and, therefore, they are easily parametrizable. Moreover $PH$ is a real vector space of dimension $(p+q)^2$ and $U(p,q)$ is an algebraic set of dimension $(p+q)^2$ (also).

Prop. If $K$ is pseudo-hermitian, then $e^{iK}$ is pseudo-unitary.

Although $dim(PH)=dim(U(p,q))$, unfortunately, the function $f:K\in PH\rightarrow e^{iK}\in U(p,q)$ is not onto; yet, I think that it is a local diffeomorphism.

Note that $SU(p,q)$ is a real algebraic set of dimension $(p+q)^2-1$.