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Description
Quaternions are a four-dimensional non-commutative algebra and a division ring of numbers introduced by Hamilton in 1843. The main obstacle in deriving eigenvalue algorithms for matrices of quaternions, due to non-commutativity, is the efficient implementation of shifts. Other linear algebra concepts naturally carry over from real or complex numbers. Reduced biquaternions are a four-dimensional commutative number algebra, introduced by Segre in 1892. The main obstacles when deriving algorithms for matrices of reduced biquaternions are the existence of non-invertible non-zero elements, and the need to consistently define some basic linear algebra concepts in this setting. We present efficient algorithms for the QR factorization, eigenvalue decomposition, and singular value decompositions of real, complex, quaternion, and reduced bi-quaternion matrices, as well as matrices of dual numbers of those four number types. The algorithms are kept as generic as possible. We present applications for computing generalized inverses and image analysis. The algorithms are efficiently implemented in Julia. This work has been partially supported by the Croatian Science Foundation under the project IP-2020-02- 2240.
