What constraints are needed to make singular value decomposition a unique transformation? While the singular value decomposition of a matrix is very general, the standard factorization of a matrix A into two singular vector matrices U and V and a singular value matrix L is not unique, in that there are often multiple choices for those matrices that all yield the original matrix A.  What set of additional conventions/constraints/normalizations is sufficient to insure that the decomposition is unique?
 A: Given the lack of answers to date, I provide what I believe to be a possible set of conventions making the SVD a unique transformation, primarily to demonstrate the question’s feasibility.


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*In general, the singular values must be treated as complex values (or signed reals), with the traditional singular values being the magnitudes of these values.

*The magnitudes of the singular values must be ordered in some specified manner, traditionally in order of decreasing magnitude.

*In the case of repeated singular values, there must be a method of uniquely resolving ties for purposes of ordering.  My approach to accomplishing this would be to define the SVD in the case of an $N{\rm{ }}\times{\rm{ }}N$ square matrix through the limiting form
$$\underline {\overline {\bf{U}} } \,\underline {\overline {\bf{\Lambda }} } \,{\underline {\overline {\bf{V}} } ^ + } = \mathop {\lim }\limits_{\varepsilon  \to 0} \left( {\underline {\overline {\bf{X}} }  + \varepsilon \left[ {\begin{array}{*{20}{c}}
N& \cdots &0\\
 \vdots & \ddots & \vdots \\
0& \cdots &1
\end{array}} \right]} \right)$$
with analogous limiting forms in the case of rectangular matrices.  [I must include the caveat that I have not exhaustively explored this approach, so there may be issues with its rigor.]

*The singular vector matrices on both sides must take the form of this factorization, with the phase matrices ${\underline {\overline {\bf{\Phi }} } _L}$ and ${\underline {\overline {\bf{\Phi }} } _R}$ being unity.  For any more arbitrary choice of singular vector matrices, it is the product ${\underline {\overline {\bf{\Phi }} } _L}\underline {\overline {\bf{\Phi }} } _R^ +$ [appropriately truncated in the case of rectangular matrices] that provides the phase terms that make the singular values complex.

*For tall matrices ($M{\rm{ }}\times{\rm{ }}N$ with $M > N$), all DOF vectors ${\underline {\bf{w}} _{Li}}$ associated with columns $i = N + 1 \to M$ of the left hand singular vector matrix must be truncated so that only the first $N$ DOF are nonzero.  For wide matrices ($M < N$), an equivalent requirement exists for the right hand singular vector matrix, and for rank deficient matrices (rank $R < \min \left( {M,N} \right)$), an equivalent requirement exists for both singular vector matrices.
