This tag is for questions regarding the Matrix Norm, a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).

A **matrix norm** of a matrix $~\lVert A \rVert~$ is any mapping from $~\mathbb R^{n×n}~$to$~\mathbb R~$ with the following three properties:

$1.~$ $~\lVert A \rVert~ = 0~$, iff $~A = 0~~~~$ **(Definiteness)**

$2.~$ $~\lVert A \rVert~ > 0~$, if $~A \ne 0~~~~$ **(Positivity)**

$3.~$ $~\lVert \alpha~A \rVert~=~|α|~\lVert A \rVert~$, for any $~α ∈ \mathbb R~~~~$ **(Homogeneity)**

$4.~$ $~\lVert A+B \rVert ≤ \lVert A \rVert + \lVert B \rVert~$ **(Triangular Inequality)**

for any matrix $~A,~ B ∈ \mathbb R^{n×n}~$.

In addition to these required properties for matrix norm, some of them also satisfy these additional properties not required of all matrix norms:

- $~\lVert A \rVert - \lVert B \rVert\le \lVert A-B \rVert~$
- $~\lVert Ax \rVert~\le~\lVert A \rVert\cdot\lVert x \rVert~~~~$
**(Subordinance)** - $~\lVert AB \rVert~\le~\lVert A \rVert\cdot\lVert B \rVert~~~~$
**(Submultiplicativity)**

A matrix norm that satisfies this additional property is called a **sub-multiplicative norm** or, **subordinate matrix norm** (in some books, the terminology matrix norm is used only for those norms which are sub-multiplicative). The set of all $~ n\times n~$ matrices, together with such a sub-multiplicative norm, is an example of a Banach algebra.

**References:**