Essays/Complete Tensor
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The complete tensor (or complete skew tensor) of order n is the array C with shape n$n such that for row i in (#:i.)n$n , the atom (<i){C is
- 0 if i is not a permutation
- 1 if i is an even permutation
- _1 if i is an odd permutation
Since the monad C.!.2 computes the parity of a permutation, the complete tensor of order n obtains by applying C.!.2 to (#:i.)n$n .
For example, for n=:3
C.!.2 ] 0 0 1 0 C.!.2 ] 0 1 2 1 C.!.2 ] 0 2 1 _1 ] i=: (#: i.) 3$3 0 0 0 0 0 1 0 0 2 0 1 0 0 1 1 0 1 2 0 2 0 0 2 1 0 2 2 1 0 0 1 0 1 1 0 2 1 1 0 1 1 1 1 1 2 1 2 0 1 2 1 1 2 2 2 0 0 2 0 1 2 0 2 2 1 0 2 1 1 2 1 2 2 2 0 2 2 1 2 2 2 t=: ,"2 ] 2 1 1": i NB. a more compact display (C.!.2 i) ; t ┌────────┬────────────┐ │ 0 0 0│ 000 001 002│ │ 0 0 1│ 010 011 012│ │ 0 _1 0│ 020 021 022│ │ │ │ │ 0 0 _1│ 100 101 102│ │ 0 0 0│ 110 111 112│ │ 1 0 0│ 120 121 122│ │ │ │ │ 0 1 0│ 200 201 202│ │_1 0 0│ 210 211 212│ │ 0 0 0│ 220 221 222│ └────────┴────────────┘ CT=: C.!.2 @ (#:i.) @ $~ CT 3 0 0 0 0 0 1 0 _1 0 0 0 _1 0 0 0 1 0 0 0 1 0 _1 0 0 0 0 0
The complete tensor can be also be represented as a sparse array.
perm=: i.@! A. i. NB. all permutations of i.n CT1=: 3 : '(C.!.2 p) (<"1 p=. perm y)} 1$.$~y' CT1 3 0 1 2 │ 1 0 2 1 │ _1 1 0 2 │ _1 1 2 0 │ 1 2 0 1 │ 1 2 1 0 │ _1 (CT -: CT1)"0 >: i.6 1 1 1 1 1 1
Cross Product
The complete tensor is used in the generalized cross product:
ip =: +/ .* cross=: [ ip CT@#@[ ip ] 'x y'=: _40 + 2 3 ?.@$ 100 x 6 15 39 y 12 14 _1 ] c=. x cross y 561 _474 96 x ip c 0 c ip x 0 y ip c 0 c ip y 0 'x y'=: _40 + 2 4 ?.@$ 100 x 6 15 39 12 y 14 _1 20 17 ] c=. x cross y 0 423 _267 339 _423 0 _66 426 267 66 0 _216 _339 _426 216 0 x ip c 0 0 0 0 c ip x 0 0 0 0 y ip c 0 0 0 0 c ip y 0 0 0 0
Determinant
] m=: _40 + 3 3 ?@$ 100 15 _35 15 33 18 10 0 _35 29 *// m 0 _17325 14355 0 _9450 7830 0 _5250 4350 0 40425 _33495 0 22050 _18270 0 12250 _10150 0 _17325 14355 0 _9450 7830 0 _5250 4350 (CT 3) * *// m 0 0 0 0 0 7830 0 5250 0 0 0 33495 0 0 0 0 0 0 0 _17325 0 0 0 0 0 0 0 +/ , (CT 3) * *// m 29250 -/ .* m 29250 det=: +/ @ , @ (CT@# * *//) det m 29250
Collected Definitions
CT =: C.!.2 @ (#:i.) @ $~ perm =: i.@! A. i. CT1 =: 3 : '(C.!.2 p) (<"1 p=. perm y)} 1$.$~y' ip =: +/ .* cross=: [ ip CT@#@[ ip ] det =: +/ @ , @ (CT@# * *//)
Contributed by Roger Hui. The complete tensor and its applications are described in the dictionary entry for C. and in the Parity & Symmetry section of J Phrases.