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.