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The Nonexistence of Pseudoquaternions in



The field of quaternions, denoted by can be represented as an isomorphic four dimensional subspace of , the space of real matrices with four rows and columns. In addition to the quaternions there is another four dimensional subspace in which is also a field and which has – in connection with the quaternions – many pleasant properties. This field is called field of pseudoquaternions . It exists in but not in . It allows to write the quaternionic linear term axb in matrix form as Mx where x is the same as the quaternion x only written as a column vector in . And M is the product of the matrix associated with the quaternion a with the matrix associated with the pseudoquaternion b . Now, the field of quaternions can also be represented as an isomorphic four dimensional subspace of over , the space of complex matrices with two rows and columns. We show that in this space pseudoquaternions with all the properties known from do not exist. However, there is a subset of for which some of the properties are still valid. By means of the Kronecker product we show that there is a matrix in which has the properties of the pseudoquaternionic matrix.



Advances in Applied Clifford AlgebrasSpringer Journals

Published: Sep 1, 2011

DOI: 10.1007/s00006-010-0273-1

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