Understanding J array

Question

Starting with something simple:

a =: 3 4 $ i.12

This creates a nice 3 x 4 matrix.

So when I try

b=: a $ i.5

I get a very long output for b. I do not understand what the output is (by the way, i.5 was arbitrary). By which I mean, I don't nderstand what $ means when it has a 2+ dimensional array as a left hand argument. Can someone explain what b is describing?

Answer 1

Here is what happens: a becomes matrix

0 1  2  3
4 5  6  7
8 9 10 11

as you just described. $ has dyadic rank of 1 _, which means it applies rows of left argument to the entire right argument and pastes result together. a $ i.5 is (0 1 2 3 $ i.5),(4 5 6 7 $ i.5),:(8 9 10 11 $ i.5)

All the subarrays have different shape, so they are coerced into least common shape that would fit them all, which in this case is 8 9 10 11. Therefore, the result has shape 3 8 9 10 11