Editing Singular value decomposition (section)

=== Hilbert–Schmidt norm{{Anchor|Hilbert–Schmidt norm|Hilbert-Schmidt norm|Hilbert–Schmidt|Hilbert-Schmidt}} ===
The singular values are related to another norm on the space of operators. Consider the [[Hilbert–Schmidt operator|Hilbert–Schmidt]] inner product on the {{tmath|n \times n}} matrices, defined by

<math display=block>
\langle \mathbf{M}, \mathbf{N} \rangle
= \operatorname{tr} \left( \mathbf{N}^*\mathbf{M} \right).
</math>

So the induced norm is

<math display=block>
\| \mathbf{M} \|
= \sqrt{\langle \mathbf{M}, \mathbf{M} \rangle}
= \sqrt{\operatorname{tr} \left( \mathbf{M}^*\mathbf{M} \right)}.
</math>

Since the trace is invariant under unitary equivalence, this shows

<math display=block>
\| \mathbf{M} \| = \sqrt{\vphantom\bigg|\sum_i \sigma_i ^2}
</math>

where {{tmath|\sigma_i}} are the singular values of {{tmath|\mathbf M.}} This is called the '''[[Frobenius norm]]''', '''Schatten 2-norm''', or '''Hilbert–Schmidt norm''' of {{tmath|\mathbf M.}} Direct calculation shows that the Frobenius norm of {{tmath|\mathbf M {{=}} (m_{ij})}} coincides with:

<math display=block>
\sqrt{\vphantom\bigg|\sum_{ij} | m_{ij} |^2}.
</math>

In addition, the Frobenius norm and the trace norm (the nuclear norm) are special cases of the [[Schatten norm]].