Inner and Outer products
This part will cover
- Inner products
- Outer products
- Matrix inverse
Recall the Rank ⍤ operator which specified what rank cells a function acts on,
M ← ? 5 5 ⍴ 10
3 2 2 9 5
9 2 10 8 2
3 4 6 5 2
9 3 5 3 2
8 8 6 8 8
(⊂⍤2)M
┌──────────┐
│3 2 2 9 5│
│9 2 10 8 2│
│3 4 6 5 2│
│9 3 5 3 2│
│8 8 6 8 8│
└──────────┘
(⊂⍤1)M
┌─────────┬──────────┬─────────┬─────────┬─────────┐
│3 2 2 9 5│9 2 10 8 2│3 4 6 5 2│9 3 5 3 2│8 8 6 8 8│
└─────────┴──────────┴─────────┴─────────┴─────────┘
(⊂⍤0)M
3 2 2 9 5
9 2 10 8 2
3 4 6 5 2
9 3 5 3 2
8 8 6 8 8
As can be seen above, a rank-n cell of a rank-r array obtained by specifying the first (r-n) indices of the array.
M[;]
3 2 2 9 5
9 2 10 8 2
3 4 6 5 2
9 3 5 3 2
8 8 6 8 8
M[1;]
3 2 2 9 5
M[1;1]
3
This specifies the cells to act on monadically; for dyadic functions, two ranks must be specified, to be interpreted as matching up cells of a certain rank from the left argument with certain cells of the right argument.
v ← ? 5 ⍴ 5
v
3 4 4 3 2
M ← 5 5 ⍴ ⍳25
M
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
⍝ See pairing of 0-cells of v and 1-cells of M
v({⍺,'-',⍵}⍤(0 1))M
3 - 1 2 3 4 5
4 - 6 7 8 9 10
4 - 11 12 13 14 15
3 - 16 17 18 19 20
2 - 21 22 23 24 25
⍝ Pairing of 1-cells of v and 1-cells of M
v({⍺,'-',⍵}⍤(1 1))M
3 4 4 3 2 - 1 2 3 4 5
3 4 4 3 2 - 6 7 8 9 10
3 4 4 3 2 - 11 12 13 14 15
3 4 4 3 2 - 16 17 18 19 20
3 4 4 3 2 - 21 22 23 24 25
N ← 5 5 ⍴ +\⍳25
N
1 3 6 10 15
21 28 36 45 55
66 78 91 105 120
136 153 171 190 210
231 253 276 300 325
⍝ Pairing of 1-cells of N and 1-cells of M
N({⍺,'-',⍵}⍤(1 1))M
1 3 6 10 15 - 1 2 3 4 5
21 28 36 45 55 - 6 7 8 9 10
66 78 91 105 120 - 11 12 13 14 15
136 153 171 190 210 - 16 17 18 19 20
231 253 276 300 325 - 21 22 23 24 25
One commonly used operation is matching every pair of elements from two vectors, one way to write this is using each and rank ⍤
v1 ← ? 5 ⍴ 5
5 5 3 4 1
v2 ← ⎕A[? 5 ⍴ 26]
HTOKY
v1(,⍤(0 1))v2
5 HTOKY
5 HTOKY
3 HTOKY
4 HTOKY
1 HTOKY
v1(,¨⍤(0 1))v2
┌───┬───┬───┬───┬───┐
│5 H│5 T│5 O│5 K│5 Y│
├───┼───┼───┼───┼───┤
│5 H│5 T│5 O│5 K│5 Y│
├───┼───┼───┼───┼───┤
│3 H│3 T│3 O│3 K│3 Y│
├───┼───┼───┼───┼───┤
│4 H│4 T│4 O│4 K│4 Y│
├───┼───┼───┼───┼───┤
│1 H│1 T│1 O│1 K│1 Y│
└───┴───┴───┴───┴───┘
There is a specific operator for this operation called the outer product (∘.f), this operator is special in that it takes a right function argument.
v1(∘.,)v2
┌───┬───┬───┬───┬───┐
│5 H│5 T│5 O│5 K│5 Y│
├───┼───┼───┼───┼───┤
│5 H│5 T│5 O│5 K│5 Y│
├───┼───┼───┼───┼───┤
│3 H│3 T│3 O│3 K│3 Y│
├───┼───┼───┼───┼───┤
│4 H│4 T│4 O│4 K│4 Y│
├───┼───┼───┼───┼───┤
│1 H│1 T│1 O│1 K│1 Y│
└───┴───┴───┴───┴───┘
'POP' 'HEAVY' 'ALT' 'SYNTH' ∘., 'ROCK' 'METAL' 'PUNK' 'WAVE'
┌─────────┬──────────┬─────────┬─────────┐
│POPROCK │POPMETAL │POPPUNK │POPWAVE │
├─────────┼──────────┼─────────┼─────────┤
│HEAVYROCK│HEAVYMETAL│HEAVYPUNK│HEAVYWAVE│
├─────────┼──────────┼─────────┼─────────┤
│ALTROCK │ALTMETAL │ALTPUNK │ALTWAVE │
├─────────┼──────────┼─────────┼─────────┤
│SYNTHROCK│SYNTHMETAL│SYNTHPUNK│SYNTHWAVE│
└─────────┴──────────┴─────────┴─────────┘
For higher rank arrays, the outer product works similarly, matching each element of both arrays
⍳2 2
┌───┬───┐
│1 1│1 2│
├───┼───┤
│2 1│2 2│
└───┴───┘
((⍳2 2)⌷¨¨⊂⊂'AB')
┌──┬──┐
│AA│AB│
├──┼──┤
│BA│BB│
└──┴──┘
(⍳2 2)∘.,((⍳2 2)⌷¨¨⊂⊂'AB')
┌──────┬──────┐
│1 1 AA│1 1 AB│
├──────┼──────┤
│1 1 BA│1 1 BB│
└──────┴──────┘
┌──────┬──────┐
│1 2 AA│1 2 AB│
├──────┼──────┤
│1 2 BA│1 2 BB│
└──────┴──────┘
┌──────┬──────┐
│2 1 AA│2 1 AB│
├──────┼──────┤
│2 1 BA│2 1 BB│
└──────┴──────┘
┌──────┬──────┐
│2 2 AA│2 2 AB│
├──────┼──────┤
│2 2 BA│2 2 BB│
└──────┴──────┘
In many applications, it is useful to reduce over the diagonal of the outer product. The inner product of vectors +.× is an example of this, which APL generalizes using the inner product f.g operator.
v1 ← ⍳5
v2 ← 'ABCDE'
v1∘./v2
┌─────┬─────┬─────┬─────┬─────┐
│A │B │C │D │E │
├─────┼─────┼─────┼─────┼─────┤
│AA │BB │CC │DD │EE │
├─────┼─────┼─────┼─────┼─────┤
│AAA │BBB │CCC │DDD │EEE │
├─────┼─────┼─────┼─────┼─────┤
│AAAA │BBBB │CCCC │DDDD │EEEE │
├─────┼─────┼─────┼─────┼─────┤
│AAAAA│BBBBB│CCCCC│DDDDD│EEEEE│
└─────┴─────┴─────┴─────┴─────┘
⍝ Diagonal
(1 1)∘⍉(v1∘./v2)
┌─┬──┬───┬────┬─────┐
│A│BB│CCC│DDDD│EEEEE│
└─┴──┴───┴────┴─────┘
⍝ The following are equivalent
,/(1 1)∘⍉(v1∘./v2)
┌───────────────┐
│ABBCCCDDDDEEEEE│
└───────────────┘
v1,./v2
┌───────────────┐
│ABBCCCDDDDEEEEE│
└───────────────┘
For arbitrary arrays, the inner product X(f.g)Y is the same as taking the outer product ∘.(f/ (g¨)) of the rows of the left argument (trailing vectors) (⊂[⍴⍴x]X) and the columns of the right argument (leading vectors) ⊂[1]Y, then removing a layer of depth ⊃⍤1. For example, the elements of X(+.×)Y are the sum of the product ∘.(+/ (ר)) of a row of X and a column of Y, which is exactly matrix multiplication.
⎕←NUM←5 5⍴0 0 0 1 2 1 1 2 3
0 0 0 1 2
1 1 2 3 0
0 0 1 2 1
1 2 3 0 0
0 1 2 1 1
⍝ The trailing vectors (rows) of NUM
⊂[⍴⍴NUM]NUM
┌─────────┬─────────┬─────────┬─────────┬─────────┐
│0 0 0 1 2│1 1 2 3 0│0 0 1 2 1│1 2 3 0 0│0 1 2 1 1│
└─────────┴─────────┴─────────┴─────────┴─────────┘
⎕←ALPHA←5 5⍴⎕A
ABCDE
FGHIJ
KLMNO
PQRST
UVWXY
⍝ The leading vectors (columns) of ALPHA
⊂[1]ALPHA
┌─────┬─────┬─────┬─────┬─────┐
│AFKPU│BGLQV│CHMRW│DINSX│EJOTY│
└─────┴─────┴─────┴─────┴─────┘
(⊂[⍴⍴NUM]NUM) (∘.(/¨)) ⊂[1]ALPHA
┌─────────────┬─────────────┬─────────────┬─────────────┬─────────────┐
│┌┬┬┬─┬──┐ │┌┬┬┬─┬──┐ │┌┬┬┬─┬──┐ │┌┬┬┬─┬──┐ │┌┬┬┬─┬──┐ │
│││││P│UU│ │││││Q│VV│ │││││R│WW│ │││││S│XX│ │││││T│YY│ │
│└┴┴┴─┴──┘ │└┴┴┴─┴──┘ │└┴┴┴─┴──┘ │└┴┴┴─┴──┘ │└┴┴┴─┴──┘ │
├─────────────┼─────────────┼─────────────┼─────────────┼─────────────┤
│┌─┬─┬──┬───┬┐│┌─┬─┬──┬───┬┐│┌─┬─┬──┬───┬┐│┌─┬─┬──┬───┬┐│┌─┬─┬──┬───┬┐│
││A│F│KK│PPP││││B│G│LL│QQQ││││C│H│MM│RRR││││D│I│NN│SSS││││E│J│OO│TTT│││
│└─┴─┴──┴───┴┘│└─┴─┴──┴───┴┘│└─┴─┴──┴───┴┘│└─┴─┴──┴───┴┘│└─┴─┴──┴───┴┘│
├─────────────┼─────────────┼─────────────┼─────────────┼─────────────┤
│┌┬┬─┬──┬─┐ │┌┬┬─┬──┬─┐ │┌┬┬─┬──┬─┐ │┌┬┬─┬──┬─┐ │┌┬┬─┬──┬─┐ │
││││K│PP│U│ ││││L│QQ│V│ ││││M│RR│W│ ││││N│SS│X│ ││││O│TT│Y│ │
│└┴┴─┴──┴─┘ │└┴┴─┴──┴─┘ │└┴┴─┴──┴─┘ │└┴┴─┴──┴─┘ │└┴┴─┴──┴─┘ │
├─────────────┼─────────────┼─────────────┼─────────────┼─────────────┤
│┌─┬──┬───┬┬┐ │┌─┬──┬───┬┬┐ │┌─┬──┬───┬┬┐ │┌─┬──┬───┬┬┐ │┌─┬──┬───┬┬┐ │
││A│FF│KKK│││ ││B│GG│LLL│││ ││C│HH│MMM│││ ││D│II│NNN│││ ││E│JJ│OOO│││ │
│└─┴──┴───┴┴┘ │└─┴──┴───┴┴┘ │└─┴──┴───┴┴┘ │└─┴──┴───┴┴┘ │└─┴──┴───┴┴┘ │
├─────────────┼─────────────┼─────────────┼─────────────┼─────────────┤
│┌┬─┬──┬─┬─┐ │┌┬─┬──┬─┬─┐ │┌┬─┬──┬─┬─┐ │┌┬─┬──┬─┬─┐ │┌┬─┬──┬─┬─┐ │
│││F│KK│P│U│ │││G│LL│Q│V│ │││H│MM│R│W│ │││I│NN│S│X│ │││J│OO│T│Y│ │
│└┴─┴──┴─┴─┘ │└┴─┴──┴─┴─┘ │└┴─┴──┴─┴─┘ │└┴─┴──┴─┴─┘ │└┴─┴──┴─┴─┘ │
└─────────────┴─────────────┴─────────────┴─────────────┴─────────────┘
⍝ Making the outer product clearer by adding labels
(' ',⊂[⍴⍴NUM]NUM),(⊂[1]ALPHA)⍪((⊂[⍴⍴NUM]NUM)(∘.(/¨))⊂[1]ALPHA)
┌─────────┬─────────────┬─────────────┬─────────────┬─────────────┬─────────────┐
│ │AFKPU │BGLQV │CHMRW │DINSX │EJOTY │
├─────────┼─────────────┼─────────────┼─────────────┼─────────────┼─────────────┤
│0 0 0 1 2│┌┬┬┬─┬──┐ │┌┬┬┬─┬──┐ │┌┬┬┬─┬──┐ │┌┬┬┬─┬──┐ │┌┬┬┬─┬──┐ │
│ │││││P│UU│ │││││Q│VV│ │││││R│WW│ │││││S│XX│ │││││T│YY│ │
│ │└┴┴┴─┴──┘ │└┴┴┴─┴──┘ │└┴┴┴─┴──┘ │└┴┴┴─┴──┘ │└┴┴┴─┴──┘ │
├─────────┼─────────────┼─────────────┼─────────────┼─────────────┼─────────────┤
│1 1 2 3 0│┌─┬─┬──┬───┬┐│┌─┬─┬──┬───┬┐│┌─┬─┬──┬───┬┐│┌─┬─┬──┬───┬┐│┌─┬─┬──┬───┬┐│
│ ││A│F│KK│PPP││││B│G│LL│QQQ││││C│H│MM│RRR││││D│I│NN│SSS││││E│J│OO│TTT│││
│ │└─┴─┴──┴───┴┘│└─┴─┴──┴───┴┘│└─┴─┴──┴───┴┘│└─┴─┴──┴───┴┘│└─┴─┴──┴───┴┘│
├─────────┼─────────────┼─────────────┼─────────────┼─────────────┼─────────────┤
│0 0 1 2 1│┌┬┬─┬──┬─┐ │┌┬┬─┬──┬─┐ │┌┬┬─┬──┬─┐ │┌┬┬─┬──┬─┐ │┌┬┬─┬──┬─┐ │
│ ││││K│PP│U│ ││││L│QQ│V│ ││││M│RR│W│ ││││N│SS│X│ ││││O│TT│Y│ │
│ │└┴┴─┴──┴─┘ │└┴┴─┴──┴─┘ │└┴┴─┴──┴─┘ │└┴┴─┴──┴─┘ │└┴┴─┴──┴─┘ │
├─────────┼─────────────┼─────────────┼─────────────┼─────────────┼─────────────┤
│1 2 3 0 0│┌─┬──┬───┬┬┐ │┌─┬──┬───┬┬┐ │┌─┬──┬───┬┬┐ │┌─┬──┬───┬┬┐ │┌─┬──┬───┬┬┐ │
│ ││A│FF│KKK│││ ││B│GG│LLL│││ ││C│HH│MMM│││ ││D│II│NNN│││ ││E│JJ│OOO│││ │
│ │└─┴──┴───┴┴┘ │└─┴──┴───┴┴┘ │└─┴──┴───┴┴┘ │└─┴──┴───┴┴┘ │└─┴──┴───┴┴┘ │
├─────────┼─────────────┼─────────────┼─────────────┼─────────────┼─────────────┤
│0 1 2 1 1│┌┬─┬──┬─┬─┐ │┌┬─┬──┬─┬─┐ │┌┬─┬──┬─┬─┐ │┌┬─┬──┬─┬─┐ │┌┬─┬──┬─┬─┐ │
│ │││F│KK│P│U│ │││G│LL│Q│V│ │││H│MM│R│W│ │││I│NN│S│X│ │││J│OO│T│Y│ │
│ │└┴─┴──┴─┴─┘ │└┴─┴──┴─┴─┘ │└┴─┴──┴─┴─┘ │└┴─┴──┴─┴─┘ │└┴─┴──┴─┴─┘ │
└─────────┴─────────────┴─────────────┴─────────────┴─────────────┴─────────────┘
⍝ Reducing over the result using catenate ,
((⊂[⍴⍴NUM]NUM)(∘.(,/(/¨)))⊂[1]ALPHA)
┌─────────┬─────────┬─────────┬─────────┬─────────┐
│┌───┐ │┌───┐ │┌───┐ │┌───┐ │┌───┐ │
││PUU│ ││QVV│ ││RWW│ ││SXX│ ││TYY│ │
│└───┘ │└───┘ │└───┘ │└───┘ │└───┘ │
├─────────┼─────────┼─────────┼─────────┼─────────┤
│┌───────┐│┌───────┐│┌───────┐│┌───────┐│┌───────┐│
││AFKKPPP│││BGLLQQQ│││CHMMRRR│││DINNSSS│││EJOOTTT││
│└───────┘│└───────┘│└───────┘│└───────┘│└───────┘│
├─────────┼─────────┼─────────┼─────────┼─────────┤
│┌────┐ │┌────┐ │┌────┐ │┌────┐ │┌────┐ │
││KPPU│ ││LQQV│ ││MRRW│ ││NSSX│ ││OTTY│ │
│└────┘ │└────┘ │└────┘ │└────┘ │└────┘ │
├─────────┼─────────┼─────────┼─────────┼─────────┤
│┌──────┐ │┌──────┐ │┌──────┐ │┌──────┐ │┌──────┐ │
││AFFKKK│ ││BGGLLL│ ││CHHMMM│ ││DIINNN│ ││EJJOOO│ │
│└──────┘ │└──────┘ │└──────┘ │└──────┘ │└──────┘ │
├─────────┼─────────┼─────────┼─────────┼─────────┤
│┌─────┐ │┌─────┐ │┌─────┐ │┌─────┐ │┌─────┐ │
││FKKPU│ ││GLLQV│ ││HMMRW│ ││INNSX│ ││JOOTY│ │
│└─────┘ │└─────┘ │└─────┘ │└─────┘ │└─────┘ │
└─────────┴─────────┴─────────┴─────────┴─────────┘
⍝ Removing a layer of depth
⊃⍤0⊢((⊂[⍴⍴NUM]NUM)(∘.(,/(/¨)))⊂[1]ALPHA)
┌───────┬───────┬───────┬───────┬───────┐
│PUU │QVV │RWW │SXX │TYY │
├───────┼───────┼───────┼───────┼───────┤
│AFKKPPP│BGLLQQQ│CHMMRRR│DINNSSS│EJOOTTT│
├───────┼───────┼───────┼───────┼───────┤
│KPPU │LQQV │MRRW │NSSX │OTTY │
├───────┼───────┼───────┼───────┼───────┤
│AFFKKK │BGGLLL │CHHMMM │DIINNN │EJJOOO │
├───────┼───────┼───────┼───────┼───────┤
│FKKPU │GLLQV │HMMRW │INNSX │JOOTY │
└───────┴───────┴───────┴───────┴───────┘
NUM,./ALPHA
┌───────┬───────┬───────┬───────┬───────┐
│PUU │QVV │RWW │SXX │TYY │
├───────┼───────┼───────┼───────┼───────┤
│AFKKPPP│BGLLQQQ│CHMMRRR│DINNSSS│EJOOTTT│
├───────┼───────┼───────┼───────┼───────┤
│KPPU │LQQV │MRRW │NSSX │OTTY │
├───────┼───────┼───────┼───────┼───────┤
│AFFKKK │BGGLLL │CHHMMM │DIINNN │EJJOOO │
├───────┼───────┼───────┼───────┼───────┤
│FKKPU │GLLQV │HMMRW │INNSX │JOOTY │
└───────┴───────┴───────┴───────┴───────┘
Using the matrix inverse ⌹ function to verify the multiplication
M ← +\ 5 5 ⍴ ⍳7
M
1 3 6 10 15
6 13 14 16 19
4 9 15 22 23
2 5 9 14 20
7 8 10 13 17
N ← +\ 5 5 ⍴ ⍳6
N
1 3 6 10 15
6 7 9 12 16
5 11 12 14 17
4 9 15 16 18
3 7 12 18 19
L ← M +.× N
L
134 285 435 560 630
275 540 789 1010 1185
290 599 891 1124 1292
193 406 615 790 895
208 423 633 820 960
L+.×⌹N
1 3 6 10 15
6 13 14 16 19
4 9 15 22 23
2 5 9 14 20
7 8 10 13 17
⌹M+.×L
1 3 6 10 15
6 7 9 12 16
5 11 12 14 17
4 9 15 16 18
3 7 12 18 19
The matrix inverse ⌹ also takes the pseudoinverse of a matrix, if the inverse does not exist, which can be used to get least squares solutions of systems of linear equations when a unique solution is not possible. Take the example of a bakery, wanting to make the most out of their ingredients
⍝ Recipes
⍝ Flour Milk Sugar Butter Eggs
Cake ← 450 0 700 500 6
Pancake ← 200 300 50 50 1
Cupcake ← 150 125 150 50 0
Cookies ← 280 0 250 200 2
Available ← 2200 1000 2200 1600 19
Since there are more ingredients than recipes, there will not be a unique solution to this problem. The system of equations here is
\[ \text{Cakes}\cdot\begin{bmatrix}450\\ 0\\ 700\\ 500\\ 2 \end{bmatrix}+\text{Pancake}\cdot\begin{bmatrix}200\\ 300\\ 50\\ 50\\ 1 \end{bmatrix}+\text{Cupcakes}\cdot\begin{bmatrix}150\\ 125\\ 150\\ 50\\ 0 \end{bmatrix}+\text{Cookies}\cdot\begin{bmatrix}280\\ 0\\ 250\\ 200\\ 2 \end{bmatrix}=\begin{bmatrix} 2200\\ 1000\\ 2200\\ 1600\\ 19 \end{bmatrix} \]
which can be written in matrix form as
\[ \begin{bmatrix} 450 & 200 & 150 & 280 \\ 0 & 300 & 125 & 0 \\ 700 & 50 & 150 & 250 \\ 500 & 50 & 50 & 200 \\ 2 & 1 & 0 & 2 \end{bmatrix} \begin{bmatrix} \text{Cakes} \\ \text{Pancake} \\ \text{Cupcake} \\ \text{Cookies} \end{bmatrix} = \begin{bmatrix} 2200 \\ 1000 \\ 2200 \\ 1600 \\ 19 \end{bmatrix} \]
which can be solved by obtaining the pseudoinverse of the matrix (represented below using the + symbol), and multiplying it by the target vector.
\[ \begin{bmatrix} \text{Cakes} \\ \text{Pancake} \\ \text{Cupcake} \\ \text{Cookies} \end{bmatrix} = \begin{bmatrix} 450 & 200 & 150 & 280 \\ 0 & 300 & 125 & 0 \\ 700 & 50 & 150 & 250 \\ 500 & 50 & 50 & 200 \\ 2 & 1 & 0 & 2 \end{bmatrix}^+ \begin{bmatrix} 2200 \\ 1000 \\ 2200 \\ 1600 \\ 19 \end{bmatrix} \]
Goods ← ⍉ ↑ Cake Pancake Cupcake Cookies
Goods
450 200 150 280
0 300 125 0
700 50 150 250
500 50 50 200
6 1 0 2
⍝ (Pseudo)inverse multiplied by the target vector
(⌹Goods)+.×Available
1.99585722 2.919959252 0.992099059 2.032347529
Then, the closest solution is baking roughly 2 cakes, 3 batches of pancakes, 1 batch of cupcakes, and 2 batches of cookies.