Transpose: Difference between revisions

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=== Dyadic usage ===
=== Dyadic usage ===
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|{{quote | "Dyadic transpose, <source lang=apl inline>x⍉y</source>, is probably one of the last primitives to be mastered for an APLer, but is actually straightforward to describe."|[[Roger Hui]]. <ref>[https://forums.dyalog.com/viewtopic.php?f=30&t=1648 ''dyadic transpose, a personal history'']. Dyalog Forums. 22 May 2020.</ref>}}
|{{quote | "Dyadic transpose, <source lang=apl inline>x⍉y</source>, is probably one of the last primitives to be mastered for an APLer, but is actually straightforward to describe."<ref>[[Roger Hui]]. [https://forums.dyalog.com/viewtopic.php?f=30&t=1648 ''dyadic transpose, a personal history'']. Dyalog Forums. 22 May 2020.</ref>}}
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For [[dyadic]] usage, the left argument X must be a [[vector]] whose length equals the [[rank]] of the right argument Y, and the elements must form a range so that <source lang=apl inline>∧/X∊⍳(1-⎕IO)+⌈/X</source> is satisfied.
For [[dyadic]] usage, the left argument X must be a [[vector]] whose length equals the [[rank]] of the right argument Y, and the elements must form a range so that <source lang=apl inline>∧/X∊⍳(1-⎕IO)+⌈/X</source> is satisfied.

Revision as of 23:21, 21 December 2021

⍉

Transpose (⍉) is an ambivalent primitive function which reorders the axes of the right argument array. The name Transpose comes from the fact that monadic application to a matrix yields its transpose. The dyadic usage is sometimes called Rearrange Axes, which better reflects the behavior of the function.

Examples

Monadic usage

Monadic Transpose reverses the axes of the right argument. When applied to a matrix, the result is its transpose. Scalar and vector arguments are unaffected. 

      3 3⍴⍳9
1 2 3
4 5 6
7 8 9
      ⍉3 3⍴⍳9  ⍝ Transpose of a matrix
1 4 7
2 5 8
3 6 9
      ⍴⍉3 4 5⍴⎕A  ⍝ The first axis goes last, the last axis comes first
5 4 3
      ⍉1 2 3  ⍝ Unaffected
1 2 3

Dyadic usage

"Dyadic transpose, x⍉y, is probably one of the last primitives to be mastered for an APLer, but is actually straightforward to describe."[1]

For dyadic usage, the left argument X must be a vector whose length equals the rank of the right argument Y, and the elements must form a range so that ∧/X∊⍳(1-⎕IO)+⌈/X is satisfied.

If all values in X are unique (X forms a permutation over the axes of Y), the axes are reordered by X so that N-th element of X specifies the new position for the N-th axis of Y. This means that, given a multi-dimensional index V of an element in the resulting array, V⌷X⍉Y corresponds to V[X]⍉Y. 

      X←3 1 2
      Y←3 4 5⍴⎕A
      ⍴X⍉Y  ⍝ First axis goes to third (X[1] is 3), second goes to first (X[2] is 1), etc.
4 5 3
      1 2 3⌷X⍉Y  ⍝ or (X⍉Y)[1;2;3]
P
      1 2 3[X]⌷Y  ⍝ or Y[3;1;2]
P

When X contains duplicates, the result has rank (1-⎕IO)+⌈/X. For the axes of Y that map to the same resulting axis, only the elements where the indices are equal over those axes are collected. This has the effect of extracting diagonal elements. If the axes are of unequal length, the resulting axis has the length of the shortest of them. This operation can be modeled as computing the resulting shape (⍴Y)⌊.+(⌊/⍬)×X∘.≠(1-⎕IO)+⍳⌈/X, then creating the array of its multi-dimensional indices ⍳, then modify each index and fetch the corresponding elements of Y {⍵[X]⌷Y}¨. 

      1 1⍉3 4⍴⎕A  ⍝ Extract diagonal from 3×4 matrix
AEI

      Y←?3 4 5 6 7⍴100
      X←3 2 3 1 2  ⍝ Left arg maps 5-dimensional Y to 3 dimensions
      ⍴X⍉Y         ⍝ Resulting shape is ⌊/¨6(4 7)(3 5)
6 4 3
      (X⍉Y)≡{⍵[X]⌷Y}¨⍳(⍴Y)⌊.+(⌊/⍬)×X∘.≠(1-⎕IO)+⍳⌈/X
1
Works in: Dyalog APL

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 ∙ Round
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 ∙ Identity (Null, Ident)
Dyadic Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator) ∙ Identity (Lev, Dex)
Quad names Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
  1. ↑ Roger Hui. dyadic transpose, a personal history. Dyalog Forums. 22 May 2020.
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