Editing Singular value decomposition (section)

== Variations and generalizations ==

=== 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.

=== Bounded operators on Hilbert spaces ===
The factorization {{tmath|\mathbf M {{=}} \mathbf U \mathbf \Sigma \mathbf V^*}} can be extended to a [[bounded operator]] {{tmath|\mathbf M}} on a separable Hilbert space {{tmath|H.}} Namely, for any bounded operator {{tmath|\mathbf M,}} there exist a [[partial isometry]] {{tmath|\mathbf U,}} a unitary {{tmath|\mathbf V,}} a measure space {{tmath|(X, \mu),}} and a non-negative measurable {{tmath|f}} such that

<math display=block>
\mathbf{M} = \mathbf{U} T_f \mathbf{V}^*
</math>

where {{tmath|T_f}} is the [[multiplication operator|multiplication by {{tmath|f}}]] on {{tmath|L^2(X, \mu).}}

This can be shown by mimicking the linear algebraic argument for the matrix case above. {{tmath|\mathbf V T_f \mathbf V^*}} is the unique positive square root of {{tmath|\mathbf M^* \mathbf M,}} as given by the [[Borel functional calculus]] for [[self-adjoint operator]]s. The reason why {{tmath|\mathbf U}} need not be unitary is that, unlike the finite-dimensional case, given an isometry {{tmath|U_1}} with nontrivial kernel, a suitable {{tmath|U_2}} may not be found such that

<math display=block>
\begin{bmatrix} U_1 \\ U_2 \end{bmatrix}
</math>

is a unitary operator.

As for matrices, the singular value factorization is equivalent to the [[polar decomposition]] for operators: we can simply write

<math display=block>
\mathbf M = \mathbf U \mathbf V^* \cdot \mathbf V T_f \mathbf V^*
</math>

and notice that {{tmath|\mathbf U \mathbf V^*}} is still a partial isometry while {{tmath|\mathbf V T_f \mathbf V^*}} is positive.

=== Singular values and compact operators ===
The notion of singular values and left/right-singular vectors can be extended to [[compact operator on Hilbert space]] as they have a discrete spectrum. If {{tmath|T}} is compact, every non-zero {{tmath|\lambda}} in its spectrum is an eigenvalue. Furthermore, a compact self-adjoint operator can be diagonalized by its eigenvectors. If {{tmath|\mathbf M}} is compact, so is {{tmath|\mathbf M^* \mathbf M}}. Applying the diagonalization result, the unitary image of its positive square root {{tmath|T_f}} has a set of orthonormal eigenvectors {{tmath| \{e_i\} }} corresponding to strictly positive eigenvalues {{tmath| \{\sigma_i\} }}. For any {{tmath|\psi}} in {{tmath|H,}}

<math display=block>
\mathbf{M} \psi = \mathbf{U} T_f \mathbf{V}^* \psi = \sum_i \left \langle \mathbf{U} T_f \mathbf{V}^* \psi, \mathbf{U} e_i \right \rangle \mathbf{U} e_i = \sum_i \sigma_i \left \langle \psi, \mathbf{V} e_i \right \rangle \mathbf{U} e_i,
</math>

where the series converges in the norm topology on {{tmath|H.}} Notice how this resembles the expression from the finite-dimensional case. {{tmath|\sigma_i}} are called the singular values of {{tmath|\mathbf M.}} {{tmath|\{\mathbf U e_i\} }} (resp. {{tmath|\{\mathbf U e_i\} }}) can be considered the left-singular (resp. right-singular) vectors of {{tmath|\mathbf M.}}

Compact operators on a Hilbert space are the closure of [[finite-rank operator]]s in the uniform operator topology. The above series expression gives an explicit such representation. An immediate consequence of this is:

:'''Theorem.''' {{tmath|\mathbf M}} is compact if and only if {{tmath|\mathbf M^* \mathbf M}} is compact.