Measure on Hermitian matrices

On the space of Hermitian matrices measure is defined as a product of all the degrees of freedom of the Hermitian matrix

(1)

Last line is due to

(2)

Overall factor will be absorbed by normalization of any average anyway so from now on you can forget it even existed. Every matrix in is in one-to-one correspondence with the vector

(3)

Elements of as a vector can be referred to with a help of double index notation . The inner product on this vector space is as always

(4)

so it is actually

(5)

Unitary conjugation leaves this inner product unchanged

(6)

so it's an orthogonal transformation in . Any orthogonal map in has . Since is connected to the identity, . The bottom line is that Jacobian of coordinate transformation is 1.