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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.
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    $\begingroup$ There are not that many normal subgroups in $GL(n,\Bbb{C})$... $\endgroup$ – abx May 24 at 3:46
  • $\begingroup$ Ah. Thank you... $\endgroup$ – Chris May 24 at 4:00