Explicit formulas for exponential of 2×2 split-complex matrices

Abstract

Split-complex (hyperbolic) numbers are ordered pairs of real numbers, written in the form $x+jy$ with $j^{2}=-1$, used to describe the geometry of the Lorentzian plane. Since a null split-complex number does not have an inverse, some methods to calculate the exponential of complex matrices are not valid for split-complex matrices. In this paper, we examined the exponential of a $2x2$ split-complex matrix in three cases : $i:~\Delta=0,~ii:~\Delta\neq0$ and $\Delta$ is not null split-complex number, $iii:~\Delta\neq0$ and $\Delta$ is a null split-complex number where $\Delta=(trA)^{2}-4detA$.