Why is `tranpose [[]] == []` in Haskell?

Question

Consider the following?

ghci> transpose []
[]
ghci> transpose [[]]
[]
ghci> transpose [[[]]]
[[[]]]
ghci> transpose [[[[]]]]
[[[[]]]]
ghci> transpose [[[[[]]]]]
[[[[[]]]]]
ghci> transpose [[[[[[]]]]]]
[[[[[[]]]]]]

One of these things is not like the other :p

It is very strange to me to see something so seemingly unelegant in base, but I suppose there's some excellent algebraic or theoretical reason why they chose to have transpose [[]] == []. Can anyone shed some light on this?

Answer 1

The type

transpose :: [[a]] -> [[a]] 

makes transpose take a list-of-lists of a. Here a is an arbitrary type, so transpose does not (and can not) inspect its input beyond two levels of lists.

Hence, the cases you posted are "seen" by transpose as follows:

ghci> transpose []
[]
ghci> transpose [[]]
[]
ghci> transpose [[something]]
[[something]]
ghci> transpose [[something]]
[[something]]
ghci> transpose [[something]]
[[something]]
ghci> transpose [[something]]
[[something]]

Now, it's easier to understand. As user2357112 already explained in their answer, the input [] is a 0*? matrix, [[]] is a 1*0 matrix, and [[something]] is a 1*1 matrix.