Editing Singular value decomposition (section)

=== Scale-invariant SVD ===

The singular values of a matrix {{tmath|\mathbf A}} are uniquely defined and are invariant with respect to left and/or right unitary transformations of  {{tmath|\mathbf A.}} In other words, the singular values of {{tmath|\mathbf U \mathbf A \mathbf V,}} for unitary matrices {{tmath|\mathbf U}} and {{tmath|\mathbf V,}} are equal to the singular values of {{tmath|\mathbf A.}} This is an important property for applications in which it is necessary to preserve Euclidean distances and invariance with respect to rotations.

The Scale-Invariant SVD, or SI-SVD,<ref>{{citation|last=Uhlmann |first=Jeffrey |author-link=Jeffrey Uhlmann |title=A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations |series=SIAM Journal on Matrix Analysis |year=2018 |volume=239 |issue=2 |pages=781–800 |url=http://faculty.missouri.edu/uhlmannj/UC-SIMAX-Final.pdf |url-status=dead |archive-url=https://web.archive.org/web/20190617095052id_/http://faculty.missouri.edu/uhlmannj/UC-SIMAX-Final.pdf |archive-date= 2019-06-17}}</ref> is analogous to the conventional SVD except that its uniquely-determined singular values are invariant with respect to diagonal transformations of {{tmath|\mathbf A.}} In other words, the singular values of {{tmath|\mathbf D \mathbf A \mathbf E,}} for invertible diagonal matrices {{tmath|\mathbf D}} and {{tmath|\mathbf E,}} are equal to the singular values of {{tmath|\mathbf A.}} This is an important property for applications for which invariance to the choice of units on variables (e.g., metric versus imperial units) is needed.