There is a fair amount of research into perturbation theory for singular value decompositions (e.g. Liu et al's 'First-Order Perturbation Analysis of Singular Vectors in Singular Value Decomposition' or Wedin's 'Perturbation bounds in connection with singular value decomposition'). These results depend on having a gap between the singular values corresponding to the singular vectors of interest, and the other singular values. I am interested in a perturbation bounds for the matrix $UV^*$ acquired from the decomposition $A=U\Sigma V^*$ when $A$ is square and nonsingular. It seems like changes in the order of the singular values shouldn't matter here, because we're summing over all the singular vectors. Are there any existing results in this direction?
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$\begingroup$ It's just about $UV^*=A|A|^{-1}$ ? $\endgroup$– Narutaka OZAWACommented Nov 30, 2021 at 2:16
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$\begingroup$ I know that $UV^*$ satisfies $UV^*=A(A^*A)^{-\frac{1}{2}}$ - it's the unitary part of the polar decomposition of $A$. I don't think it satisfies $UV^*=A|A|^{-1}$ if $|A|$ is the determinant. I could get a perturbation bound by in terms of the condition number of $A$ and a linear approximation for $A \mapsto (A^*A)^{-\frac{1}{2}}$, using the perturbation theory for $A\mathbf{x}=\mathbf{b}$, but I'd been hoping there'd be something better. $\endgroup$– user7868Commented Nov 30, 2021 at 3:16
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