What is an intuitive way to understand Haar measure as defined for random matrices, say, $N\times N$ orthogonal or unitary matrices?


My understanding for what Haar measure means for $U(1)$ is that it can be thought of as a measure over a uniform distribution of phases on a circle, i.e. a matrix representing $M \in U(1)$ can be parameterized with an angle $\theta$ so that $$d\mu(M) = \frac {d\theta}{2\pi} $$


What is a correct generalization of this intuition to $N>1$? In particular, are there any explicit parameterizations of Haar measure that resemble writing down angles that are uniformly distributed?


One possibility that came to mind is that eigenvectors, rows and/or columns have (generalized) phases that can be thought of as direction angles that are in some sense uniform over $SO(N)$ or $U(N)$? (Edit: seems like this would not be the case for rows and columns.)


Another possibility I have thought of is using Givens rotations to parameterize an orthogonal matrix using the resulting rotation angles that bring it to the identity matrix. Are there any known results about the distribution of the Givens angles? It would seem plausible that they could be uniformly distributed, but given that the Givens rotations are usually applied in a particular fashion to achieve diagonalization, that could introduce correlations that would result in nonuniformity.


(Caveat: I'm new to all this so I could very well be wrong about even trying to conceptualize such a question.)



Intuition for Haar measure of random matrix
You want to think of the Haar measure $d\mu(U)$ as a way of measuring uniformity in the group $U(N)$ of unitary $N\times N$ matrices. 


To form your intuition, consider $N=1$. You then have $U=e^{i\phi}$, with $0<\phi\leq 2\pi$ and $d\mu(U)=d\phi$ measures the perimeter of the unit circle. This is a uniform measure, because $d(\phi+\phi\_0)=d\phi$ for any fixed phase shift $\phi\_0$. You could write the requirement of uniformity in the form $d\mu(UU\_0)=d\mu(U)$, with $U\_0=e^{i\phi\_0}$ the unitary matrix corresponding to the phase shift $\phi\_0$.


Once your intuition is formed for $N=1$, you simply generalize to $N>1$ using the same definition of uniformity, $d\mu(UU\_0)=d\mu(U)$ for any fixed $U\_0\in U(N)$. For orthogonal (or symplectic) matrices you use the same definition of uniformity, with $U\_0$ now restricted to the orthogonal or symplectic subgroup of $U(N).$


To explicitly write down the Haar measure $d\mu(U)$ in terms of the matrix elements of $U$ is only easily done for a few small values of $N$. (In particular, there is no relationship to random directions of rows or columns, as Yemon Choi pointed out.) You typically do not need such explicit expressions, since integrals with the Haar measure can be evaluated by using only the definition of uniformity.




---


In response to the follow-up question: If you wish to evaluate Haar-measure integrals of polynomials of matrix elements of $U$, you can use the socalled Weingarten functions.


<http://en.wikipedia.org/wiki/Weingarten_function>


Here is a Mathematica program to generate these,


<http://arxiv.org/abs/1109.4244>


If you need an explicit expression for the Haar measure, the steps to take are the following:


1) parameterize your matrix $U$ in terms of a set of real parameters $\{x\_i\}$.


2) calculate the metric tensor $m\_{ij}$, defined by $\sum\_{ij}|dU\_{ij}|^2 = \sum\_{ij}m\_{ij}dx\_i dx\_j$


3) obtain the Haar measure by equating $d\mu(U) = ($Det $m)^{1/2}\prod\_i dx\_i$


This is the general recipe. In practice, for many parameterizations the answer is in the literature. In particular, for the Haar measure in Euler angle parameterizations see:


<http://arxiv.org/abs/math-ph/0205016>


<http://www.cft.edu.pl/~karol/pdf/ZK94.pdf>


