Within $GL(n,\mathbb C)$ let $H$ denote the normal subgroup generated by products of positive-definite, Hermitian matrices. What is this subgroup? And what is the quotient $GL(n,\mathbb C)/H$? Do these groups have names? Some of the more obvious facts:
- When $n=1$, $H$ is isomorphic to the multiplicative group $\mathbb R^+$ and $GL(1,\mathbb C)/H$ is isomorphic to the circle group $\mathbb S^1$.
- Polar decomposition guarantees that every matrix is equivalent mod $H$ to a unitary matrix.
- Any element of $H$ has positive, real determinant.