I am reading the Dynamical approach to the random matrix from Horong Tzer Yau. My question is about how to define a probability measure on real symmetric or complex Hermitian matrices.
In his book, he said we could obtain the probability distributions on real symmetric or complex Hermitian matrices by defining a density function directly on the set of matrices:
I just don't understand what does the "dH" mean? And how does it relate to the probability measure on real symmetric or complex Hermitian matrices. Hope someone could help, thanks!
This is shorthand for saying that the integral of the left hand side over any measurable set is equal to the integral of the right hand side over any measurable set. These integrals are each taken with respect to the measure $dH$. In the real symmetric case, $dH$ is the real Lebesgue measure on $\mathbb{R}^{n(n+1)/2}$, where we interpret each component of this measure as one of the (nonstrict) upper triangular entries of a symmetric matrix. In the complex Hermitian case, it is similar, except the diagonal is identified with $\mathbb{R}^n$ and the strict upper triangular part is identified with $\mathbb{C}^{n(n-1)/2}$.