Sure, but . . . the space of NxN Hermitian matrices is noncompact, so using Haar measure seems likely to result in a space of infinite measure. A method I think would work might be to use the standard normal distribution on the real vector space of dimension 2N^2 (total measure = 1), and then use the usual restriction-of-a-normal-distribution-to-a-subspace to define the measure on the (real dim N^2) subspace of Hermitian matrices. But maybe there's something simpler. --Dan On 2012-11-21, at 8:32 PM, Andy Latto wrote:
On Wed, Nov 21, 2012 at 9:14 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Homework question: Is there a more-or-less obvious unique natural probability measure on the space of NxN complex Hermitian matrices?
There's a canonical left-translation-invariant measure, up to a constant factor, on any topological group, called Haar measure. For a compact Lie group, it's also right translation invariant, and the measure of the whole group is finite, so you can normalize to make the measure of the whole group 1.
Andy
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