# Application of Quaternionic Matrices for Finite Turns’ Sequence Representation in Space

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Victor Kravets, Tamila Kravets, Olexiy Burov (2017). Application of Quaternionic Matrices for Finite Turns’ Sequence Representation in SpaceMechanics, Materials Science & Engineering, Vol 9. doi:10.2412/mmse.17.56.743

Authors: Victor Kravets, Tamila Kravets, Olexiy Burov

ABSTRACT. With the use of mathematical apparatus of monomial (1,0,-1)-matrices-(4×4), the Rodrigues’ vector formula describing the finite turn, is represented with quaternionic matrices. Two ways to deduct the Rodrigues’ formula in quaternionic matrices are provided: with the use of basic (1,0,-1)-matrices-(4×4), equivalent to the quaternion and conjugate quaternion; the vector and the opposite vector. It is demonstrated that the commutative product of two matrices-(4×4) equivalent to the quaternion, determines the finite turn matrix, and the commutative product of two matrices-(4×4) equivalent to the conjugate quaternion determines the inverse matrix of the finite turn if the Rodrigues-Hamilton parameters are taken as quaternion components. With this use of mathematical induction method, the obtained results are generalized for the cases of arbitrary sequence of finite independent turns in space. The offered formula for representation of finite turns in space with quaternionic matrices is distinguished by its compactness and convenience for both analytic treatments and efficient computational algorithms development.

Keywords: monomial (1,0,-1)-matrices-(4×4), quaternionic matrices, Rodrigues’ formula, parameters of Rodrigues-Hamilton, finite rotation

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 Victor Kravets, Tamila Kravets, Olexiy Burov (2017). Identities of Vector Algebra as Associative Properties of Multiplicative Compositions of Quaternion MatricesMechanics, Materials Science & Engineering, Vol 8. doi:10.2412/mmse.47.87.900

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