Editing Matrix norm (section)

==Preliminaries==
Given a [[field (mathematics)|field]] <math>\ K\ </math> of either [[real number|real]] or [[complex number]]s (or any complete subset thereof), let <math>\ K^{m \times n}\ </math> be the {{mvar|K}}-[[vector space]] of matrices with <math>m</math> rows and <math>n</math> columns and entries in the field <math>\ K ~.</math> A matrix norm is a [[Norm (mathematics)|norm]] on <math>\ K^{m \times n}~.</math>

Norms are often expressed with [[double vertical bar]]s (like so: <math>\ \|A\|\ </math>).  Thus, the matrix norm is a [[Function (mathematics)|function]] <math>\ \|\cdot\| : K^{m \times n} \to \R^{0+}\ </math> that must satisfy the following properties:<ref name=":0">{{cite web |last=Weisstein |first=Eric W. |title=Matrix norm |website=mathworld.wolfram.com |url=https://mathworld.wolfram.com/MatrixNorm.html |access-date=2020-08-24 |lang=en }}</ref><ref name=":1">{{Cite web |title=Matrix norms |website=fourier.eng.hmc.edu |url=http://fourier.eng.hmc.edu/e161/lectures/algebra/node12.html | access-date=2020-08-24 }}</ref>

For all scalars <math>\ \alpha \in K\ </math> and matrices <math>\ A, B \in K^{m \times n}\ ,</math>
* <math> \|A\| \ge 0\ </math> (''positive-valued'')
* <math> \|A\| = 0 \iff A=0_{m,n}</math> (''definite'')
* <math> \left\| \alpha\ A \right\| = \left| \alpha \right|\ \left\|A\right\|\ </math> (''absolutely homogeneous'')
* <math> \| A + B \| \le \| A \| + \| B \|\ </math> (''sub-additive'' or satisfying the ''triangle inequality'')

The only feature distinguishing matrices from rearranged vectors is [[matrix multiplication|multiplication]]. Matrix norms are particularly useful if they are also '''sub-multiplicative''':<ref name=":0"/><ref name=":1"/><ref>{{cite journal |last=Malek-Shahmirzadi |first=Massoud |year=1983 |title=A characterization of certain classes of matrix norms |journal=Linear and Multilinear Algebra |volume=13 |issue=2 |pages=97–99 | doi=10.1080/03081088308817508| issn=0308-1087 |lang=en }}</ref>

* <math>\ \left\| AB \right\| \le \left\| A \right\| \left\| B \right\|\ </math>{{efn|group=Note|
The condition only applies when the product is defined, such as the case of [[Square matrix|square matrices]] (<math>\ m = n\ </math>). More generally, multiplication of the matrices must be possible: <math>\ A \in K^{\ell \times m}\ </math> and <math>\ B \in K^{m \times n} ~;</math> further, the two norms <math>\ \|A\|\ </math> and <math>\ \|B\|\ </math> must either have the same definitions, only differing in the matrix dimensions, or two different types of norms that are none the less "consistent" (see below).
}}

Every norm on <math>\ K^{n\times n}\ </math> can be rescaled to be sub-multiplicative; in some books, the terminology ''matrix norm'' is reserved for sub-multiplicative norms.<ref>{{cite book |last=Horn |first=Roger A. |year=2012 |title=Matrix analysis |edition=2nd |publisher=Cambridge University Press |location=Cambridge, UK |others=Johnson, Charles R. |isbn=978-1-139-77600-4 |oclc=817236655 |pages=340–341 }}</ref>