# Distribution of unitary matrices generated using SVD of a random matrix in the Ginibre ensemble

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

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).