Research on the Computation and Properties of the Hyperbolic Complex Determinant

In this paper, the calculation results and properties of determinants under hyperbolic complex numbers are studied. The hyperbolic complex number is a commutative ring consisting of two real numbers with zero divisors. This paper overcomes the difficulty that the hyperbolic complex number contains zero divisors, and calculates the decomposition of the determinant of the hyperbolic complex number by its decomposition property. This paper investigates the decomposition operation of hyperbolic complex determinants and proves that any hyperbolic complex determinant can be decomposed into two real determinants through zero factors. Based on this decomposition, the necessary and sufficient conditions for the invertibility of hyperbolic complex determinants are further derived. In addition, the property that the determinant of the product of hyperbolic complex matrices is equal to the product of their respective determinants is discussed. This will lay a foundation for the study of hyperbolic complex numbers in algebraic theory and provide a new direction for their application in physics.