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Let $A \in \mathbb C^{n\times n}$.

Let $A$ be in the Ginibre ensemble, meaning each element of $A$ is independently drawn from the normal distribution $\mathcal{N}(\mu = 0, \sigma = 1)$.

Let $A = U\Sigma V^*$ be its singular value decomposition, where $U$, $V$ are unitary and $\Sigma$ is a diagonal matrix.

What is the distribution of $UV^*$? Is it uniform?


Note: If we use the QR decomposition instead, $A = QR$, then $Q$ is a uniformly random unitary matrix. See: http://home.lu.lv/~sd20008/papers/essays/Random%20unitary%20%5Bpaper%5D.pdf

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It's uniform, for a similar reason as for QR. For $A\in GL_n(\mathbb C)$ define

$$f(A)=A(A^*A)^{-1/2}\in U_n(\mathbb C)$$ using the square root and inverse of a positive definite matrix. Note that $f(U\Sigma V^*)=UV^*.$ For any unitary matrix $U,$ the notes you linked to shows that $UA$ has the same distribution as $A.$ We have

$$f(UA)=UA(A^*A)^{-1/2}=Uf(A)$$

so $Uf(A)$ has the same distribution as $f(A).$ But the only left-invariant probability measure on $U_n(\mathbb C)$ is the uniform measure (Haar's theorem).

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