Quaternion Function Reference

abs

Absolute value, or modulus
(Quaternion and octonion overloadings of standard MATLAB® function)

Syntax

Y = abs(X)

Description

abs(X) returns an array Y such that each element of Y is the absolute value, or modulus, of the corresponding element of X.

If X is a complex quaternion, abs(X) returns the complex modulus, the square root of the complex semi-norm, computed in the same way (the square root of the sum of the squares of the components of the quaternion). The semi-norm of a complex quaternion can vanish (see references), and therefore so can the modulus.

Examples

abs(quaternion(1,1,1,1))

ans = 2
abs is vectorized, and hence can operate on arrays: 
abs([qi, qj, qk, qi + qj])

ans =

    1.0000    1.0000    1.0000    1.4142
It can also operate on complex quaternions, with complex modulus, in general: 
abs(1 + i + qi + qj + qk)

ans = 1.8174 + 0.5503i
The following shows that a complex quaternion can have a vanishing modulus (and semi-norm): 
abs(i + qi)

ans = 0

See Also

MATLAB® function: abs
QTFM functions: normq, normo

References

  1. Sangwine, S. J. and Alfsmann, D., 'Determination of the biquaternion divisors of zero, including the idempotents and nilpotents', Advances in Applied Clifford Algebras, 20, (2), May 2010, 401–410, DOI: 10.1007/s00006-010-0202-3, also available as e-print arXiv:0812.1102, 8 December 2008, available at http://arxiv.org/abs/arxiv:0812.1102.
  2. W. R. Hamilton, Lectures on Quaternions, Lecture VII, §672, p669. Hodges and Smith, Dublin, 1853. Available online at: http://historical.library.cornell.edu/math/.

© 2008-2022 Stephen J. Sangwine and Nicolas Le Bihan

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