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Polar Decomposition of Complexified Quaternions and Octonions

Abstract

We present a hitherto unknown polar representation of complexified quaternions (also known as biquaternions), also applicable to complexified octonions. The complexified quaternion is factored into the product of two exponentials, one trigonometric or circular, and one hyperbolic. The trigonometric exponential is a real quaternion, the hyperbolic exponential has a real scalar part and imaginary vector part. This factorisation is shown to be isomorphic to the polar decomposition of linear algebra.

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Notes

  1. The terms biquaternion and complexified quaternion are synonymous here and mean a quaternion with coefficients in \({\mathbb {C}}\).

  2. The term hypercomplex means a generalization of the complex numbers based on roots of \(\pm 1\), and with dimension 2, 4, 8, etc. It is of course possible for this formula to work for other types of root than hypercomplex numbers (for example, matrices).

  3. We thank one of the referees for pointing out the orthogonality of \({\varvec{\alpha }}\) and \({\varvec{\beta }}\) in the example.

  4. The transpose of this matrix is also valid, one or other must be chosen by convention.

  5. An alternative with the Hermitian transpose on the right is also possible, corresponding to the alternative ordering of the factors in Theorem 1.

  6. This statement has the following equivalent in matrix form: \((PQ)^T=Q^T P^T\).

  7. This method is specific to the biquaternion case, and is not applicable to the polar decomposition of matrices in general.

  8. The matrix representation of a biquaternion q can be computed using the function call adjoint(q, ’real’).

  9. This is a slight but common abuse of notation — the quaternion and octonion \(\varvec{i},\varvec{j},\varvec{k} \) should be regarded as distinct.

  10. This choice follows from the way the octonions are implemented in [11] as a pair of quaternions using the Cayley–Dickson construction:

    $$\begin{aligned} (w + x\varvec{i} + y\varvec{j} + z\varvec{k}) + (a + b\varvec{i} + c\varvec{j} + d\varvec{k})\varvec{l} = w + x\varvec{i} + y\varvec{j} + z\varvec{k} + a\varvec{l} + b\varvec{m} + c\varvec{n} + d\varvec{o}. \end{aligned}$$

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Acknowledgements

We thank the two referees who reviewed this paper very thoroughly and made some excellent suggestions for minor improvements, which have, we believe, improved the paper considerably.

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Correspondence to Stephen J. Sangwine.

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Communicated by Leo Dorst.

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Sangwine, S.J., Hitzer, E. Polar Decomposition of Complexified Quaternions and Octonions. Adv. Appl. Clifford Algebras 30, 23 (2020). https://doi.org/10.1007/s00006-020-1048-y

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  • DOI: https://doi.org/10.1007/s00006-020-1048-y

Keywords

  • Biquaternion
  • Quaternion
  • Polar decomposition

Mathematics Subject Classification

  • Primary 11R52
  • Secondary 15A23