Consimilarity of hybrid number matrices and hybrid number matrix equations $\mathrm{A}\tilde{\mathrm{X}}-\mathrm{X}\mathrm{B}=\mathrm{C}$

This research aims to investigate the consimilarity of hybrid number matrices and to develop solutions for matrix equations associated with these numbers. Hybrid numbers are an innovative algebraic structure that unifies dual, complex, and hyperbolic (perplex) number systems. These numbers are isomorphic to split quaternions and hold significant importance in mathematical theory and physical applications, especially in the context of non-commutative algebraic structures. The paper demonstrates how linear matrix equations associated with hybrid numbers, such as $\mathbf{A}\stackrel{\mathbf{~}}{\mathbf{X}}\mathbf{-}\mathbf{X}\mathbf{B}=\mathbf{C}$, can be solved by reducing them to Sylvester equations through real matrix representations. The concept of consimilarity, defined as a transformation preserving structural properties of matrices without requiring invertibility, is thoroughly examined. This concept is applied to analyze eigenvalues, diagonalization, and both linear and nonlinear matrix equations involving hybrid number matrices. By investigating the consimilarity of hybrid number matrices, the study introduces new algebraic methods and computational techniques, expanding classical results in matrix theory to hybrid numbers. This research not only advances theoretical insights into hybrid number systems but also opens avenues for practical applications in scientific and engineering fields.