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

Abstract

The field of quaternions, denoted by $${\mathbb{H}}$$ can be represented as an isomorphic four dimensional subspace of $${\mathbb{R}^{4\times 4}}$$ , the space of real matrices with four rows and columns. In addition to the quaternions there is another four dimensional subspace in $${\mathbb{R}^{4\times 4}}$$ 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 $${\mathbb{R}^{4\times 4}}$$ but not in $${\mathbb{H}}$$ . 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 $${\mathbb{R}^4}$$ . 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 $${\mathbb{C}^{2\times 2}}$$ over $${\mathbb{R}}$$ , the space of complex matrices with two rows and columns. We show that in this space pseudoquaternions with all the properties known from $${\mathbb{R}^{4\times 4}}$$ do not exist. However, there is a subset of $${\mathbb{C}^{2\times 2}}$$ for which some of the properties are still valid. By means of the Kronecker product we show that there is a matrix in $${\mathbb{C}^{4\times 4}}$$ which has the properties of the pseudoquaternionic matrix.