Definition:Norm/Matrix Space

Definition

Let $\map {\MM_\GF} {m, n}$ denote the vector space of matrices of order $m \times n$ over a field $\GF$.

A norm over $\map {\MM_\GF} {m, n}$ is known as a matrix norm.

A matrix norm on $\map {\MM_\GF} {m, n}$ is a map from $\map {\MM_\GF} {m, n}$ to the nonnegative reals:

$\norm {\, \cdot \,}: \map {\MM_\GF} {m, n} \to \R_{\ge 0}$

$(\text N 1)$	$:$	Positive Definiteness:	$\ds \forall \mathbf A \in \map {\MM_\GF} {m, n}:$	$\ds \norm {\mathbf A} = 0 $	$\ds \iff $	$\ds \mathbf A = \mathbf 0_{m, n} $	where $\mathbf 0_{m, n}$ denotes the zero matrix of order $m \times n$
$(\text N 2)$	$:$	Positive Homogeneity:	$\ds \forall x \in \map {\MM_\GF} {m, n}, \lambda \in \GF:$	$\ds \norm {\lambda \mathbf A} $	$\ds = $	$\ds \norm \lambda \times \norm {\mathbf A} $	where $\norm \lambda$ denotes the (division ring) norm of $\lambda$
$(\text N 3)$	$:$	Triangle Inequality:	$\ds \forall \mathbf A, \mathbf B \in \map {\MM_\GF} {m, n}:$	$\ds \norm {\mathbf A + \mathbf B} $	$\ds \le $	$\ds \norm {\mathbf A} + \norm {\mathbf B} $

A matrix norm can also be styled as norm of matrix.

\((\text N 1)\)	$:$	Positive Definiteness:	\(\ds \forall \mathbf A \in \map {\MM_\GF} {m, n}:\)	\(\ds \norm {\mathbf A} = 0 \)	\(\ds \iff \)	\(\ds \mathbf A = \mathbf 0_{m, n} \)	where $\mathbf 0_{m, n}$ denotes the zero matrix of order $m \times n$
\((\text N 2)\)	$:$	Positive Homogeneity:	\(\ds \forall x \in \map {\MM_\GF} {m, n}, \lambda \in \GF:\)	\(\ds \norm {\lambda \mathbf A} \)	\(\ds = \)	\(\ds \norm \lambda \times \norm {\mathbf A} \)	where $\norm \lambda$ denotes the (division ring) norm of $\lambda$
\((\text N 3)\)	$:$	Triangle Inequality:	\(\ds \forall \mathbf A, \mathbf B \in \map {\MM_\GF} {m, n}:\)	\(\ds \norm {\mathbf A + \mathbf B} \)	\(\ds \le \)	\(\ds \norm {\mathbf A} + \norm {\mathbf B} \)