Editing Singular value decomposition (section)

==Norms==

=== Ky Fan norms ===
The sum of the {{tmath|k}} largest singular values of {{tmath|\mathbf M}} is a [[matrix norm]], the [[Ky Fan]] {{tmath|k}}-norm of {{tmath|\mathbf M.}}<ref>{{Cite journal|last=Fan|first=Ky.|date=1951|title=Maximum properties and inequalities for the eigenvalues of completely continuous operators|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=37|issue=11|pages=760–766|doi=10.1073/pnas.37.11.760|pmid=16578416|pmc=1063464|bibcode=1951PNAS...37..760F|doi-access=free}}</ref>

The first of the Ky Fan norms, the Ky Fan 1-norm, is the same as the [[operator norm]] of {{tmath|\mathbf M}} as a linear operator with respect to the Euclidean norms of {{tmath|K^m}} and {{tmath|K^n.}} In other words, the Ky Fan 1-norm is the operator norm induced by the standard <math>\ell^2</math> Euclidean inner product. For this reason, it is also called the operator 2-norm. One can easily verify the relationship between the Ky Fan 1-norm and singular values. It is true in general, for a bounded operator {{tmath|\mathbf M}} on (possibly infinite-dimensional) Hilbert spaces

<math display=block>
\| \mathbf M \| = \| \mathbf M^* \mathbf M \|^\frac{1}{2}
</math>

But, in the matrix case, {{tmath|(\mathbf M^* \mathbf M)^{1/2} }} is a [[normal matrix]], so <math> \|\mathbf M^* \mathbf M\|^{1/2} </math> is the largest eigenvalue of {{tmath|(\mathbf M^* \mathbf M)^{1/2},}} i.e. the largest singular value of {{tmath|\mathbf M.}}

The last of the Ky Fan norms, the sum of all singular values, is the [[trace class|trace norm]] (also known as the 'nuclear norm'), defined by <math>\| \mathbf M \| = \operatorname{Tr}(\mathbf M^* \mathbf M)^{1/2}</math> (the eigenvalues of {{tmath|\mathbf M^* \mathbf M}} are the squares of the singular values).

=== 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]].