Explicit Solutions of Linear Discrete Delay Systems: A Comprehensive Survey of Delayed Matrix Functions and Their Applications

This survey provides a systematic review of delayed matrix functions and their role in deriving explicit solutions for linear discrete delay systems. Tracing the evolution from foundational single-delay first-order systems to sophisticated multi-delay configurations, we cover delayed matrix exponentials, sines, and cosines for both commutative and non-commutative coefficient matrices, as well as generalizations for two-sided delay terms. We synthesize closed-form solution representations for a wide spectrum of initial value problems and highlight applications across stability analysis, controllability, iterative learning control, and finite-time stability. The paper concludes with a critical discussion identifying open problems-including extensions to higher-order differences, Volterra-type systems, and non-commutative multi-delay scenarios-serving as a unified reference connecting algebraic construction to analytical utility in linear discrete dynamical systems with memory. The review is reported in accordance with the PRISMA 2020 guidelines; the completed checklist and flow diagram are provided.