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The Moore–Penrose Inverse and Singular Value Decomposition of Split Quaternions

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Abstract

Using the concept of a transposition anti-involution in Clifford algebra \(C \, \ell _{1,1}\) and the isomorphisms \({\mathbb {H}}_s \cong C \, \ell _{1,1} \cong \text {Mat}(2,{\mathbb {R}}),\) where \({\mathbb {H}}_s\) is the algebra of split quaternions and \(\text {Mat}(2,{\mathbb {R}})\) is the algebra of \(2 \times 2\) real matrices, one can find the Moore–Penrose inverse \(q^{+}\) of a non-zero non-invertible split quaternion q. In particular, using a well-known algorithm for finding the Moore-Penrose inverse \(Q^{+}\) of a non-zero \(2 \times 2\) matrix Q of rank 1, one can give four governing equations that the (unique) split quaternion \(q^{+}\) corresponding to \(Q^{+}\) must satisfy. We show how a dyadic expansion and a Singular Value Decomposition can be found for any split quaternion q, and we relate them to \(q^{+}\). Results presented in this paper may be useful in a plethora of recent applications of split quaternions.

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Change history

  • 02 July 2021

    The original version of this article was revised due to a broken link in the reference.

Notes

  1. The involutive anti-automorphism \(\tau \) of \({\mathbb {H}}_s\) in (3) is a special case of a transposition anti-automorphism \(T_\varepsilon \tilde{}\) of the (universal) Clifford algebra \(C \, \ell _{p,q}\) [4,5,6]. On basis monomials, \(T_\varepsilon \tilde{}(e_{i_1}e_{i_2}\cdots e_{i_s})=(e_{i_1}e_{i_2}\cdots e_{i_s})^{-1}\) where \(\{e_1,\ldots ,e_{p+q}\}\) is an orthonormal basis in \({\mathbb {R}}^{p,q}\) and \(i_1< i_2< \cdots < i_s\). Monomials \((e_{i_1}e_{i_2}\cdots e_{i_s})^{-1}\) together with the algebra unity provide the so called reciprocal basis for \(C \, \ell _{p,q}\). See also [18, Page 5] for a right-left symmetry of group rings K[G] which gives an anti-automorphism \(\star :K[G] \rightarrow K[G]\) analogous to \(T_\varepsilon \tilde{}.\)

  2. We mention explicitly the isomorphism \(\psi \) because all computations that follow have been performed and verified with a Maple package CLIFFORD [2, 3]. The grade involution is also often denoted as \({\hat{a}}\) while the reversion is often denoted as \({\tilde{a}}\) for any element a in \(C \, \ell _{p,q}\) [14].

  3. We restrict our attention to real matrices due to the isomorphism \({\mathbb {H}}_s \cong \text {Mat}(2,{\mathbb {R}}).\)

  4. Our approach via the spinor representation realized by \(2 \times 2\) matrices is superior to trying to use a regular \(4 \times 4\) representation of \({\mathbb {H}}_s\) because it reduces to just these two cases: \({{\,\mathrm{\mathrm {rank}}\,}}A= 1 \, \text{ or }\, 2.\)

  5. This follows from a well-known fact (cf. [12]) that when Q is as \(m \times n\) matrix, then \({{\,\mathrm{\mathrm {rank}}\,}}Q = {{\,\mathrm{\mathrm {rank}}\,}}Q^TQ = {{\,\mathrm{\mathrm {rank}}\,}}QQ^T\). So, when \(Q=\varphi (q)\) has rank 1, matrices \(Q^TQ\) and \(QQ^T\) have the same characteristic polynomial with two distinct roots: one zero and one nonzero. Thus, the characteristic polynomial \(\Delta \) is also minimal and, of course, it is of degree 2. It can be found by a hand computation or with a command climinpoly in CLIFFORD. For an algorithm to compute coefficients of the characteristic polynomial for any \(u \in C \, \ell _{p,q}\) by the method of Leverrier, see [10].

  6. Equality \(c=I_q^2\) also follows from the fact that \(I_q\) is the constant term in a characteristic polynomial of \(q \in {\mathbb {H}}_s,\) namely, \(\Delta _q(t) =t^2-2q_0t+(q_0^2+q_1^2-q_2^2-q_3^2)\). This, of course, is to be expected since the constant term in the characteristic polynomial of \(Q=\varphi (q)\) is \(\det Q = I_q\).

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Acknowledgements

The author is very grateful to three anonymous reviewers for their careful reading of the manuscript and helpful comments on how to improve the presentation.

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  1. Sarasota, FL, 34238, USA

    Rafał Abłamowicz

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Communicated by Jacques Helmstetter.

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The original version of this article was revised due to a broken link in the reference.

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Abłamowicz, R. The Moore–Penrose Inverse and Singular Value Decomposition of Split Quaternions. Adv. Appl. Clifford Algebras 30, 33 (2020). https://doi.org/10.1007/s00006-020-01058-8

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