Why are matrix norms defined the way they are? Given $A$ a square matrix
Define:
$\|A\|_1$ as the max absolute column sum
$\|A\|_2$ as the sum of the squares of each element 
$\|A\|_\infty$ as the max absolute row sum
Pray tell, why are matrix norms defined this way? Is this a property inherited or derived from the vector norms? 
(Source doesn't say: http://www.personal.soton.ac.uk/jav/soton/HELM/workbooks/workbook_30/30_4_matrx_norms.pdf )
 A: Matrices can be considered as linear operators. And for a linear operator $A:X\to Y$, where $X,Y$ are normed spaces with norms $\|.\|_X,\|.\|_Y$, the definition of the operator norm is 
$$\|A\|=\sup\limits_{x\in X,x\neq 0}{\frac{\|Ax\|_Y}{\|x\|_X}}$$
If you use this definition, then the obtained matrix norm is called induced norm, because it is induced from the vector norms of the underlying vector spaces $X$ and $Y$. Such norms naturally satisfy also the last norm property $\|AB\|\leq \|A\|\|B\|$. But this property is not a real property of the norm (there are only $3$ properties), it is just that some authors use the terminology a matrix norm, only for those norms which satisfy this additional property (see Wikipedia). For example, if you have square matrix $A\in \mathbb R^{n\times n}$, $A:(\mathbb R^n,l^2)\to (\mathbb R^n,l^2)$, where $(\mathbb R^n,l^2)$ means the vector space $\mathbb R^n$ equipped with the $l^2$ Euclidean norm, the resulting induced matrix norm is 
$$\|A\|_2=\sup\limits_{x\neq 0}{\frac{\|Ax\|_2}{\|x\|_2}}=\sup\limits_{x\neq 0}{\frac{\sqrt{\langle Ax,Ax\rangle}}{\sqrt{\langle x,x\rangle}}}=\sup\limits_{x\neq 0}{\sqrt{\frac{\langle A^TAx,x\rangle}{\langle x,x\rangle}}}=\sqrt{\lambda_{\max}(A^TA)}$$
You also can find the derivation of the $\|A\|_1,\|A\|_\infty$ and other induced $\|A\|_p$ norms.
