Split-complex Number - Matrix Representations

Matrix Representations

One can easily represent split-complex numbers by matrices. The split-complex number

z = x + j y

can be represented by the matrix

Addition and multiplication of split-complex numbers are then given by matrix addition and multiplication. The modulus of z is given by the determinant of the corresponding matrix. In this representation, split-complex conjugation corresponds to multiplying on both sides by the matrix

For any real number a, a hyperbolic rotation by a hyperbolic angle a corresponds to multiplication by the matrix

The diagonal basis for the split-complex number plane can be invoked by using an ordered pair (x,y) for and making the mapping

Now the quadratic form is Furthermore,

so the two parametrized hyperbolas are brought into correspondence. The action of hyperbolic versor then corresponds under this linear transformation to a squeeze mapping

The commutative diagram interpretation of this correspondence has A = B = {split-complex number plane}, C = D = R2, f is the action of a hyperbolic versor, g & h are the linear transformation by the matrix of ones, and k is the squeeze mapping.

Note that in the context of 2 × 2 real matrices there are in fact a great number of different representations of split-complex numbers. The above diagonal representation represents the jordan canonical form of the matrix representation of the split-complex numbers. For a split-complex number z = (x,y) given by the following matrix representation:

its Jordan canonical form is given by:

where and,

Thus all the "different" matrix representations of the split-complex numbers are in fact equivalent up to similarity to the jordan normal form. The determinant, trace and eigenvalues (not eigenvectors) remain unchanged under similarity transformations.