The Nonexistence of Pseudoquaternions in $${\mathbb{C}^{2\times 2}}$$

Abstract

The field of quaternions, denoted by H can be represented as an isomorphic four dimensional subspace of R 4×4 , the space of real matrices with four rows and columns. In addition to the quaternions there is another four dimensional subspace in R 4×4 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 R 4×4 but not in H. 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 R 4 . 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 C 2×2 over R, the space of complex matrices with two rows and columns. We show that in this space pseudoquaternions with all the properties known from R 4×4 do not exist. However, there is a subset of C 2×2 for which some of the properties are still valid. By means of the Kronecker product we show that there is a matrix in C 4×4 which has the properties of the pseudoquaternionic matrix.Mathematics Subject Classification (2010). 11R52, 12E15.