Editing Singular value decomposition (section)

=== 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.