Condense

Z←<R    or    Z←<[X] R converts R to a Hypercomplex array if (⍴R)[X]∊2 4 8 or a Real array if (⍴R)[X]=1.

X is an optional numeric singleton axis with X∊⍳⍴⍴R &mddash; otherwise (including if R is a scalar), an AXIS ERROR is signalled. If [X] is omitted, the operation applies to the last axis.

R is an arbitrary Real numeric array, i.e., Fixed-Precision Boolean, Integer, Floating Point, Arithmetic Progression Array, or Multiple-Precision Integer/Rational or Floating Point — otherwise, a DOMAIN ERROR is signalled. The X-axis length (⍴R)[X] must be 1, 2, 4, or 8 — otherwise, a LENGTH ERROR is signalled.

Z is the corresponding Real or Hypercomplex array of shape (X≠⍳⍴⍴R)/⍴R using the numbers along the X-axis of R as the coefficients of the resulting Real or Hypercomplex array. If (⍴R)[X] is 1, the result is an array of Real numbers, (⍴R)[X] is 2, the result is an array of Complex numbers, (⍴R)[X] is 4, the result is an array of Quaternion numbers, (⍴R)[X] is 8, the result is an array of Octonion numbers.

For example,

      <23 23       <10 20 10J20       <2 4⍴⍳8 1i2j3k4 5i6j7k8       <[1] 2 4⍴⍳8 1J5 2J6 3J7 4J8       <2 8⍴(⍳8),⌽⍳8 1i2j3k4l5ij6jk7kl8 8i7j6k5l4ij3jk2kl1       <[1] 2 8⍴(⍳8),⌽⍳8 1J8 2J7 3J6 4J5 5J4 6J3 7J2 8J1       ⍴⎕←<2 3 1⍴⍳6 1 2 3 4 5 6 2 3       ⍴⎕←<2 3 2⍴(⍳6),⌽⍳6 1J2 3J4 5J6 6J5 4J3 2J1 2 3       ⍴⎕←<[1] 2 3 2⍴(⍳6),⌽⍳6 1J6 2J5 3J4 4J3 5J2 6J1 3 2       ⍴⎕←<2 3 4⍴(⍳12),⌽⍳12  1i2j3k4   5i6j7k8 9i10j11k12 12i11j10k9 8i7j6k5 4i3j2k1 2 3       ⍴⎕←<2 3 8⍴(⍳24),⌽⍳24  1i2j3k4l5ij6jk7kl8         9i10j11k12l13ij14jk15kl16 17i18j19k20l21ij22jk23kl24 24i23j22k21l20ij19jk18kl17 16i15j14k13l12ij11jk10kl9   8i7j6k5l4ij3jk2kl1 2 3

Identities

R ←→ < > R   for all Real (1==R) R (see Dilate for the definition of monadic Right Caret)
R ←→ <[X] >[X] R   for all Real (1==R) R
R ←→ > < R   for all R with (¯1↑⍴ R)∊1 2 4 8
1/R ←→ > < R   for all R with (¯1↑⍴1/R)∊1 2 4 8
R ←→ >[X] <[X] R   for all non-scalar R with (⍴R)[X]∊1 2 4 8

Acknowledgements

This symbol and its name were suggested by David A. Rabenhorst.