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The Moore-Penrose inverse and singular value decomposition of split quaternions. (English) Zbl 1444.15004

Unlike the more familiar algebra of (Hamiltonian) quaternions, the algebra of split quaternions is not a division algebra whence the notion of a Moore-Penrose inverse makes sense. The author defines it axiomatically, proves existence and uniqueness and provides an algorithm for its computation. He then proceeds with singular value decomposition and dyadic expansion (Kronecker product expansion) for non-invertible and invertible split quaternions and relations between these concepts.
There is a well-known isomorphism between the algebra of split quaternions and the algebra of real two-by-two matrices which would allow a transfer of Moore-Penrose inverses, singular value decomposition and dyadic expansion from the matrix setting. Nonetheless, the purely quaternionic formulations of this paper are of interest as they unveil more of the non-trivial underlying structure. Moreover, they prepare the ground for similar investigations in high-dimensional Clifford algebras where matrix models are no longer computationally efficient and potentially even more obscure with regard to algebraic structures.

MSC:

15A09 Theory of matrix inversion and generalized inverses
15A18 Eigenvalues, singular values, and eigenvectors
15A66 Clifford algebras, spinors
15A23 Factorization of matrices
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