I know there are different definitions of Matrix Norm, but I want to use the definition on WolframMathWorld, and Wikipedia also gives a similar definition.
The definition states as below:
Given a square complex or real $n\times n$ matrix $A$, a matrix norm $\|A\|$ is a nonnegative number associated with $A$ having the properties
1.$\|A\|>0$ when $A\neq0$ and $\|A\|=0$ iff $A=0$,
2.$\|kA\|=|k|\|A\|$ for any scalar $k$,
3.$\|A+B\|\leq\|A\|+\|B\|$, for $n \times n$ matrix $B$
Then, as the website states, we have $\|A\|\geq|\lambda|$, here $\lambda$ is an eigenvalue of $A$. I don't know how to prove it, by using just these four properties.