# What is the normal subgroup generated by nonsingular, positive definite matrices? [closed]

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.
New contributor
Chris is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
• There are not that many normal subgroups in $GL(n,\Bbb{C})$... – abx May 24 at 3:46
• Ah. Thank you... – Chris May 24 at 4:00