Editing Matrix norm (section)

==Schatten norms==
{{Further|Schatten norm}}

The Schatten ''p''-norms arise when applying the ''p''-norm to the vector of [[singular value decomposition|singular values]] of a matrix.<ref name=":1" /> If the singular values of the <math>m \times n</math> matrix <math>A</math> are denoted by ''&sigma;<sub>i</sub>'', then the Schatten ''p''-norm is defined by

:<math> \|A\|_p = \left( \sum_{i=1}^{\min\{m,n\}} \sigma_i^p(A) \right)^{1/p}.</math>

These norms again share the notation with the induced and entry-wise ''p''-norms, but they are different.

All Schatten norms are sub-multiplicative. They are also unitarily invariant, which means that <math>\|A\| = \|UAV\|</math> for all matrices <math>A</math> and all [[unitary matrix|unitary matrices]] <math>U</math> and <math>V</math>.

The most familiar cases are ''p'' = 1, 2, &infin;. The case ''p'' = 2 yields the Frobenius norm, introduced before. The case ''p'' =&nbsp;&infin; yields the spectral norm, which is the operator norm induced by the vector 2-norm (see above). Finally, ''p'' = 1 yields the '''nuclear norm''' (also known as the ''trace norm'', or the [[Singular Value Decomposition#Ky Fan norms|Ky Fan]] 'n'-norm<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|pmc=1063464|pmid=16578416|bibcode=1951PNAS...37..760F|doi-access=free}}</ref>), defined as:

: <math>\|A\|_{*} = \operatorname{trace} \left(\sqrt{A^*A}\right) = \sum_{i=1}^{\min\{m,n\}} \sigma_i(A),</math>

where <math>\sqrt{A^*A}</math> denotes a positive semidefinite matrix <math>B</math> such that <math>BB=A^*A</math>. More precisely, since <math>A^*A</math> is a [[positive semidefinite matrix]], its [[square root of a matrix|square root]] is well defined. The nuclear norm <math>\|A\|_{*}</math> is a [[convex envelope]] of the rank function <math>\text{rank}(A)</math>, so it is often used in [[mathematical optimization]] to search for low-rank matrices.

Combining [[von Neumann's trace inequality]] with [[Hölder's inequality]] for Euclidean space yields a version of [[Hölder's inequality]] for Schatten norms for <math> 1/p + 1/q = 1 </math>:

: <math>\left|\operatorname{trace}(A^*B)\right| \le \|A\|_p \|B\|_q,</math>

In particular, this implies the Schatten norm inequality

: <math> \|A\|_F^2 \le \|A\|_p \|A\|_q. </math>