What constraints are needed to make singular value decomposition a unique transformation?

Question

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?

Answer 1

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.

1. 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.

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

3. 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.]

4. 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.

5. 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.