[ssfrr]

it's an unfortunate terminology collision.

- array languages: rank is the dimensionality of an array, i.e. a vector is rank-1, a matrix is rank-2, a N-D array is rank-N

- linear algebra: rank is the number of linearly-independent columns (and also rows)

So for example, if you have a 5x5 matrix where 4 of the columns are linearly independent, it would be rank-4 in the linear algebra sense, and rank-2 in the array language sense.

I guess (though I've never really thought of it before) that you could say that the array-language definition is the rank (in the linear algebra sense) of the index space. Not sure if that's intentional.

[juliangoldsmith]

Rank in the array language terminology seems to be synonymous with tensor order (sometimes called tensor rank). Ken Iverson (APL's creator) was a mathematician, so I'm not that surprised to learn the term came from a branch of math.