When I try to derive the formula for $\| \mathbf{A} \|_1$ from the general definition of a matrix norm (section "Matrix norms induced by vector norms"), I get $\sum_{i = 1}^{M} \sum_{j = 1}^{N} |a_{i,j}|$?
But my textbook and the same wiki page says that it should be $\max_{1 \leq j \leq M} \sum_{i = 1}^{N} |a_{i,j}|$. How can you arrive at the latter from the definition of matrix norms and the definition of the vector norm $\| \circ \|_1$?
On
Same Notation for Different Norms.
The Wikipedia source uses this notation
$$\|A\|_1 = \sup\{\|Ax\|_1:\|x\|=1\}$$
This is called the operator norm when both domain and range carry the $1$-norm. Under this notation it can be shown $\|A\|_1 =$ the largest column sum.
The http://fourier.eng.hmc.edu source uses the definition $$\|A\|_p =p\text{-norm of }A \text{ if you treat is at a vector of length = #entries}.$$
The second source has $\|A\|_1 = \sum_{i,j} |a_{ij}|$ by definition.
These are different objects. For example consider the $d \times d$ identity matrix. The operator norm is $1$ but in the second notation the norm is $d$.
\begin{aligned} \|A\|_1 &=\max_{x\ne0}\frac{\|Ax\|_1}{\|x\|_1}\\ &=\max_{x\ne0}\frac{\left\|A\sum_j x_je_j\right\|_1}{\|x\|_1}\\ &=\max_{x\ne0}\frac{\left\|\sum_j x_jAe_j\right\|_1}{\|x\|_1}\\ &\le\max_{x\ne0}\frac{\sum_j |x_j|\|Ae_j\|_1}{\|x\|_1}\\ &\le\max_{x\ne0}\frac{\left(\max_j\|Ae_j\|_1\right)\sum_j |x_j|}{\|x\|_1}\\ &=\max_{x\ne0}\frac{\left(\max_j\|Ae_j\|_1\right)\|x\|_1}{\|x\|_1}\\ &=\max_j\|Ae_j\|_1\\ &=\max_j\sum_i|a_{ij}|. \end{aligned}