Interpret this expression $\left \| A \right \| _1 = \max \limits _{1 \leq j \leq n} \sum _{i=1} ^m | a_{ij} |$

Question

The "$1$" norm of a matrix (in some cases, which cases?), according to Wikipedia, is defined as follows:

$$\left \| A \right \| _1 = \max \limits _{1 \leq j \leq n} \sum _{i=1} ^m | a_{ij} |$$

Then there's an example of the matrix

$$ A = \begin{bmatrix} -3 & 5 & 7 \\ 2 & 6 & 4 \\ 0 & 2 & 8 \\ \end{bmatrix}, $$

and its "$1$" norm is calculated as follows

$$||A||_1 = max(|−3|+2+0, 5+6+2, 7+4+8) = max(5,13,19) = 19$$

which suprises me, because the way I read $\left \| A \right \| _1 = \max \limits _{1 \leq j \leq n} \sum _{i=1} ^m | a_{ij} |$ is different, and is the following

Sum the maximum elements in absolute value of each row from row $i$ to $m$

and this interpretation would lead me to the result $|7| + |6| + |8| = 21$.

What makes mathematicians think that the notation above does not mean how I interpret it? How would you interpret it personally?

Answer 1

$$ \max _{1 \leq j \leq n} \sum _{i=1} ^m | a_{ij} | $$

should be read as follows:

$$ \max_{1 \leq j \leq n} \left(\sum _{i=1} ^m | a_{ij} |\right) $$.

That is, for each fixed $j$, you first evaluate $\sum _{i=1} ^m | a_{ij} |$. For a fixed $j$, $\sum _{i=1} ^m | a_{ij} |$ is the sum of the absolute value of the elements in the $j$th column. Once you have computed all of these sums, you take the maximum one.

Essentially, your confusion comes because you misinterpreted the order of the operations.

Answer 2

The interpretation given is correct. The max is taken on the sum and not the other way around. Your interpretation would be written as:

$$ \sum_{i=1}^m \max_{1 \leq j \leq n} | a_{ij} | $$

Hope that helps!