This analysis demonstrates the connections between matrix exponential and Sylvester matrix equations, highlighting structural patterns in recursive operations.
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.
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Mersin et al. (2025) studied this question.
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