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A function $n3{cdot}:mathcal M_n(mathbb C)longrightarrowmathbb R$ is a emph{matrix norm}, if, for all $A,Binmathcal M_n(mathbb C)$ it satisfies the following five axioms:

begin{description}[itemsep=-0.3em]
item[(1)label{nonn}] $n3{A}geq0$ hfill nonnegative
item[(1a)label{pos}] $n3{A}=0 ; Longleftrightarrow ; A=0$ hfill positive
item[(2)label{hom}] $n3{c,A}=|c|n3{A} quad forall cinmathbb C$ hfill homogeneous
item[(3)label{tr}] $n3{A+B}leq n3{A}+n3{B}$ hfill triangle inequality
item[(4)label{sub}] $n3{A,B}leq n3{A}n3{B}$ hfill submultiplicativity
end{description}

The first four properties of a matrix norm are identical to the axioms for a verctor norm. A norm on matrices that does not satisfy property eqref{sub} for all $A$ and $B$ is a emph{vector norm on matrices}, sometimes called a emph{generalized matrix norm}. The notions of a matrix seminorm and a generalized matrix seminorm may also be defined via omission of axiom eqref{pos}.

The first four properties of a matrix norm are identical to the axioms for a verctor norm. A norm on matrices that does not satisfy property ref{sub} for all $A$ and $B$ is a emph{vector norm on matrices}, sometimes called a emph{generalized matrix norm}. The notions of a matrix seminorm and a generalized matrix seminorm may also be defined via omission of axiom ref{pos}.

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