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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 $N>1$? Is it possible that eigenvectors, rows and columns have (generalized) phases that can be thought of as direction angles that are in some sense uniform over $SO(N)$ or $U(N)$?

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.)

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.)

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 $N>1$? Is it possible that eigenvectors, rows and columns have (generalized) phases that can be thought of as direction angles that are in some sense uniform over $SO(N)$ or $U(N)$?

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

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 $N>1$? Is it possible that eigenvectors, rows and columns have (generalized) phases that can be thought of as direction angles that are in some sense uniform over $SO(N)$ or $U(N)$?

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.)

What is an intuitive way to understand Haar measure as defined for, say, $N\times N$ orthogonal matrices? Is it correct to think of it as having matrices with eigenvectors, rows and columns pointing in any random direction in $SO(N)$, generalizing the notion of uniform distribution of phases on a circle?

Perhaps phrased more precisely, does the distribution of the direction angles of eigenvectors become in some sense uniform over $SO(N)$?

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 $N>1$? Is it possible that eigenvectors, rows and columns have (generalized) phases that can be thought of as direction angles that are in some sense uniform over $SO(N)$ or $U(N)$?

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