How can I perform a scalar product on a matrix of pairs?

Question

I have a matrix Mcontaining pairs (i.e. arrays of size 2). Given a pair p, how can I get a 2D matrix M'containing the result of the scalar product of p and each element of M?

    (0, 0) (0, 1)                             8 5
M = (1, 0) (1, 1)       p = (2, 2)       M' = 5 2
    (2, 0) (2, 1)                             4 1

M and p are defined :

M =. 3 2 2 $ 0 0 0 1 1 0 1 1 2 0 2 1
p =. 2 2

I have an implementation of scalar product in J :

sp =. +/ @: *: @: -

It works on pairs :

    0 0 sp p
8
    0 1 sp p
5

But not on the full matrix, because of bad length :

    p sp M
|length error: sp
|   p     sp M

How should I deal with that ? This is probably easy for J geniuses, but I'm just a poor newbie.

Answer 1

Use

p sp"1 M
 8 5
 5 2
 4 1

You want sp here to read first-rank cells of M:

<"1 M
┌───┬───┐
│0 0│0 1│
├───┼───┤
│1 0│1 1│
├───┼───┤
│2 0│2 1│
└───┴───┘

so sp"1 will work on each cell (p sp 0 0, p sp 0 1, etc). sp (infinite rank) tries to apply sp once for the whole matrix M.

Likewise sp"2 will apply sp on second-rank cells of M:

<"2 M
 ┌───┬───┬───┐
 │0 0│1 0│2 0│
 │0 1│1 1│2 1│
 └───┴───┴───┘

so

p sp"2 M
 8 5
 2 5
 0 5

(the pairs: p sp 2 2 $ 0 0 0 1, p sp 2 2 $ 1 0 1 1 and p sp 2 2 $ 2 0 2 1)