Laplace Theorem for hyperbolic complex determinants

Abstract

This paper studies the Laplace theorem for hyperbolic complex determinants, aiming to explore the algebraic properties of hyperbolic complex numbers and their applications in analysis. The research significance lies in laying the foundation for the algebraic theory of hyperbolic complex analysis and promoting research development in physics and related fields. The research methods include defining hyperbolic complex numbers and their operational rules, decomposing hyperbolic complex determinants into two real determinants, and further proving the Laplace theorem for hyperbolic complex matrices. The research results indicate that hyperbolic complex determinants possess unique decomposition properties, and the Laplace theorem is applicable to hyperbolic complex matrices, providing an effective method for calculating high-order determinants and partitioned matrix determinants. The discussion section emphasizes the theoretical innovations of this research, such as proposing new definitions of hyperbolic complex algebraic structures, as well as methodological innovations, such as providing a concise method for calculating high-order determinants. Furthermore, it anticipates the potential application value of the research results in fields such as physics and hyperbolic complex analysis.