Quaternion Function Reference

associator

Associator

Syntax

A = associator(X, Y, Z, option)

Description

This function is provided for octonions only, since quaternion multiplication is associative and the function would return either zero or one, depending on the option in the quaternion case, regardless of the data parameters.

A = associator(X, Y, Z) returns (X .* Y) .* Z - X .* (Y .* Z). The same result is returned with option 'diff'.

A = associator(X, Y, Z, 'prod') returns that octonion that multiplies the product (X .* Y) .* Z on the right to give the result of the product X .* (Y .* Z)

Examples

Octonions are not associative, but quaternion-like subsets are, as shown below (this shows the difference associator, which is zero when the three parameters associate):

>> associator(oi, oj, ok)
 
ans = 0 * I + 0 * J + 0 * K + 0 * L + 0 * M + 0 * N + 0 * O
 
>> associator(oi, ol, oo)
 
ans = 0 * I + 2 * J + 0 * K + 0 * L + 0 * M + 0 * N + 0 * O  
Three octonions chosen at random have two evaluation orders for their product in a given order, and the associator multiplies one of these results to give the other:
>> x = rando; y = rando; z = rando;
>> ((x .* y) .* z) .* associator(x, y, z, 'prod')

ans =

0.2046 - 0.3172 * I + 0.3426 * J - 0.2111 * K + 0.4448 * L - 0.423 * M + 0.005592 * N - 0.5646 * O

>> x .* (y .* z)

ans =

0.2046 - 0.3172 * I + 0.3426 * J - 0.2111 * K + 0.4448 * L - 0.423 * M + 0.005592 * N - 0.5646 * O

See Also

QTFM function: commutator

References

  1. Richard D. Schafer, 'An Introduction to Non-Associative Algebras', Academic Press, 1966. Page 13.
  2. S. J. Sangwine, `Octonion associators', ArXiv e-print 1509.07718, http://arxiv.org/abs/1509.07718, 25 September 2015.

(c) 2008-2016 Stephen J. Sangwine and Nicolas Le Bihan

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