Questions tagged [matrix-norms]

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:

https://en.wikipedia.org/wiki/Matrix_norm

http://mathworld.wolfram.com/MatrixNorm.html