I'm describing a column wise operation in a paper, but I'm unsure about what is the correct notation.
$c[i] = \text{max}(C[:, i])$
The equation above is supposed to mean that each dimension $i$ of the vector $c$ is equal to the maximum value among all rows in the column $i$ of matrix $C$.
Is it clear or is there a better way to describe the operation?
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$\begingroup$This looks very much like “programmer notation” to me. That is not necessarily a bad thing if that is your audience.
For a more “mathematical” way of expressing this, you would proceed like this:
First, for some reason it is common not to subscript a matrix directly, but to introduce new names for it’s entries: $C = (c_{ij})_{1 \leq i \leq m, 1 \leq j \leq n}$ (where $m \times n$ is the type of the matrix). If you use the lower case version of the letter you used to name the matrix (like I did), you will probably be understood even if you don’t do this. You can also use $C_{ij}$ for the entries but you should mention it explicitly (unless it is common in your field).
Second, since we now used $c$ for the entries of the matrix, we will have to use another letter for your vector $c$. Pick one, I will use $d$.
Third, using $C[:,i]$ to denote the $i$-th column is very unusual. The normal (though arguably less elegant) way is to express the “loop” explicitly. Also, conventionally, $i$ is used for row indices, $j$ is used for columns (but that is not very important). This results in something like $$d_j = \max_{1 \leq i \leq m} c_{ij}.$$
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