Let $A$ be an $m \times n$ matrix, and $a_{ij}$ a matrix entry.
Usually the index $i$ indicates the row and $j$ the column of the matrix.
So changing the index $i$ makes you move in the $vertical$ direction and changing the index $j$ makes you move in the $horizontal$ direction through the matrix.
I noticed that this is exactly the other way around for how we use coordinates in the cartesian-plane, where we have a point in the plane $p = (x,y)$, and by changing the first entry we move in the $horizontal$ diretion and by changing the second entry we move in the $vertical$ direction.
I know that in a sense they are just arbitrary agreements so we can communicate about it, but I thought that maybe someone has some deeper inside about why this is the case or is it just a coincidence? And wouldn't you agree that changing $i$ to be the column and $j$ to be the row be more natural?
I'm not sure of the reason of the difference, but I agree that unifying it may be better.
Indeed, this is quite confusing in some applications like when you are trying to plot a matrix as heatmap vs showing a matrix as digital image, the first index (row index) will become the negative y coordinate on image plane, and column index becomes the positive x coordinate... It's easy to implement this wrong and get the image upsidedown flipped or transposed. This is caused by the different convention of the two fields...