[ 1 2 ]
[ 3 4 ]
This represents some linear transformation, and is a matrix. Are you trying to argue otherwise?FYI, matrices transform between vectors within a vector space.
You bring up a good point though, if this were meant to be a transformation, then we're talking about modules (Z/2^8Z being the underlying ring) and not vector spaces, which is fine. I was needlessly narrow when I said "vector spaces" earlier.
Why do you mention things like data structures, enums and labels? This is math, not C++.
Anyway, the point is, any arbitrary 2D arrangement of numbers can be a valid matrix, whatever those numbers may represent.
I'm genuinely claiming they aren't numbers due to the explicit context. The entries are colors with numerals as labels.
Now, you've said something slightly different where I think we can agree. If the entries are from a set closed under a notion of multiplication and addition, then it represents a matrix even if it isn't being used in that way. But I still say that if you're merely using the array as a place to keep some data then I won't be using the word matrix, I'll just call it an array.
Anyway. I started this by saying it was interesting that we use the word matrix slightly differently. I still think that's interesting and think it's totally fine if you want to call an array a matrix even if the entries don't come from something where a transformation makes sense. I wouldn't "correct" someone's usage. I just think it's interesting how we use it different. Anyway, I feel like you just think I'm dumb, so I'm ending this here. You might not, but it's hard to read the vibe via text. If you have something to add you think will get me to change my word choices, feel free to respond and I'll read it, but I'm not replying on this thread anymore.
Square ones do, but m x n ones represent linear maps from an n-dimensional to m-dimensional vector space (over the/a field containing the elements of the matrix).