On the Commutators and Anti-Commutators of Matrix Functions Involving Exchange Matrices

In this study, we focus on the commutators and anti-commutators of the matrix exponential, trigonometric, and hyperbolic functions of the nth order exchange matrix Ψ with an arbitrary matrix Λ. We also explore their behaviour under recursive commutator and anti-commutator operations to highlight structural patterns and interactions between such matrices. Moreover, we emphasize the connection of these results with Sylvester matrix equations, which provides both theoretical insights and potential applications.