Let $A$ be a $n \times n$ non-singular matrix. Then we can take many alternative SVD from from $A$. Like $A = U_1 \Sigma V_1^\top = U_2 \Sigma V_2^\top = \cdots$ where $\Sigma$ is an ordered diagonal (invertible) matrix of singular values. Can we say that $U_1V_1^\top = U_2V_2^\top = \cdots$ ? I.e., I am curious about the uniqueness of $UV^T$.
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2$UV^T$ is the unitary factor in the polar decomposition of $A$. It is uniquely determined by $A(A^TA)^{-1/2}$ when $A$ is non-singular. – user1551 Feb 21 '23 at 08:23
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@user1551 Thank you for your perfect reply! – bsw1907 Feb 21 '23 at 08:37
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Possibly related – Rodrigo de Azevedo Feb 21 '23 at 09:19
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See also this post – Ben Grossmann Feb 22 '23 at 04:23
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Thanks to everyone who commented! – bsw1907 Feb 25 '23 at 10:32