Inner Product

Inner Product (.), is a dyadic operator, which will produce a dyadic function when applied with two dyadic functions. In APL, the inner product is a generalisation of the matrix product, which allows not only addition-multiplication, but any dyadic functions given.

Examples

      x ← 1 2 3
      y ← 4 5 6
      x ,.(⊂,) y ⍝ visualizing of pairing
┌─────────────┐
│┌───┬───┬───┐│
││1 4│2 5│3 6││
│└───┴───┴───┘│
└─────────────┘
      x {⊂⍺,'+',⍵}.{⊂⍺,'×',⍵} y ⍝ visualizing function application in matrix multiplication
┌───────────────────────────┐
│┌─────────────────────────┐│
││┌─────┬─┬───────────────┐││
│││1 × 4│+│┌─────┬─┬─────┐│││
│││     │ ││2 × 5│+│3 × 6││││
│││     │ │└─────┴─┴─────┘│││
││└─────┴─┴───────────────┘││
│└─────────────────────────┘│
└───────────────────────────┘
      x+.×y ⍝ matrix multiplication
32

Note that the shapes of the arguments must be compatible with each other: The last axis of the left argument must have the same length as the first axis of the right argument, or formally, for X f.g Y it must be that (¯1↑⍴X)≡(1↑⍴Y). The shape of the result is (¯1↓⍴X),(1↓⍴Y).

For example, when applying inner product on two matrices, the number of columns in the left array must match with number of rows in the right array, otherwise we will get an error.

      ⎕  ← x ← 2 3⍴⍳10
1 2 3
4 5 6
      ⎕ ← y ← 4 2⍴⍳10
1 2
3 4
5 6
7 8
      x+.×y 
LENGTH ERROR
      x+.×y
        ∧
      ⎕ ← y ← 3 2⍴⍳10 ⍝ reshape y to be compatible with x
      x+.×y
22 28
49 64

External links

Documentation

APL built-ins [edit]
Primitives (Timeline)	Functions
Scalar
		Monadic	Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root
		Dyadic	Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
Non-Scalar
		Structural	Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
		Selection	First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
		Selector	Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
		Computational	Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
Operators	Monadic	Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis
Operators	Dyadic	Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator)
Quad names	Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector

Inner Product

Examples

External links

Documentation

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