NB. Some matrix techniques expressed in the J
NB. programming notation.

NB. Identity

I =: =/~ i. 3

NB. Dot Product

dp =: +/ @: *

NB. Tensor Product

tp =: dyad def '((3 1 & $) x.) (+/ . *) (1 3 & $) y.'

NB. Cross Product

wx =: monad def '3 3 $ 0 1 _1 _1 0 1 1 _1 0 * 0 2 1 2 0 0 1 0 0 { y.'
xub =: adverb def '3 3 $ 0 1 _1 _1 0 1 1 _1 0 * 0 2 1 2 0 0 1 0 0 { m.'

x =: adverb def '(wx m.) & (+/ . *)'
x =: adverb def 'm. xub & (+/ . *)'
x =: adverb def '(3 3 $ 0 1 _1 _1 0 1 1 _1 0 * 0 2 1 2 0 0 1 0 0 { m.) & (+/ . *)'

NB. Identities

NB. (a +/ . * I) -: a

a =: +/~ i. 3

NB. (u +/ . * (v tp w)) -: (u dp v)*w

'u v w' =: I
'u v w' =: 3 3 $ 1 2 3 4 5 6 7 8 9

NB. (u +/ . * w xub) -: - w x u

NB. Translation

T =: monad def '(I,.0) , y. , 1'

NB. Rotation
NB. L - rotation axis line
NB. w - unit vector parallel to L
NB. Q - point on L
NB. o - angle of rotation

R =: dyad def '((2 o. y.)*I) + ((1 - 2 o. y.) * x. tp x.) + (1 o. y.)* wx x.'
R =: dyad def '((2 o. y.)*I) + ((1 - 2 o. y.) * x. tp x.) + (1 o. y.)* x. xub'

rot =: conjunction def '((((2 o. n.)*I) + ((1 - 2 o. n.) * m. tp m.) + (1 o. n.)* m. xub),.0),(y. - y. +/ . * ((2 o. n.)*I) + ((1 - 2 o. n.) * m. tp m.) + (1 o. n.)* m. xub),1'

rot =: conjunction define
r =. ((2 o. n.)*I) + ((1 - 2 o. n.) * m. tp m.) + (1 o. n.)* m. xub
(r,.0),(y. - y. +/ . * r),1
)

NB. Example:

(1 0 0 rot (o. 1r2)) 0 0 0
(0 1 0 rot (o. 1r2)) 0 0 0
(0 0 1 rot (o. 1r2)) 0 0 0

NB. Scaling
NB. Q - scaling origin
NB. c - scaling factor
NB. w - unit vector parallel to scaling direction

NB. Uniform scaling

S =: dyad def '((y.*I),.0) , ((1-y.)*x.) , 1'

NB. Nonuniform scaling