Representation of solutions to continuous and discrete first-order linear matrix equations with pure delay involving nonpermutable matrices

Abstract: This paper presents a unified treatment of first-order linear matrix equations (FLMEs) with pure delay in both continuous and discrete time, under the general setting of noncommutative coefficient matrices. In the continuous-time framework, we consider delayed matrix differential equations of the form [ \dot{X}(\vartheta) = A_0 X(\vartheta-\sigma) + X(\vartheta-\sigma) A_1 + G(\vartheta), \quad \vartheta \geq 0, ] where (A_0, A_1 \in \mathbb{R}^{d \times d}) satisfy (A_0 A_1 \neq A_1 A_0), (G(\vartheta)) is a prescribed matrix function, and (\sigma >0) denotes the delay. We derive explicit representations of the solution for initial data (X(\vartheta) = \Psi(\vartheta)), (\vartheta \in [-\sigma,0]), highlighting the structural impact of noncommutativity. For the discrete-time analogue, the system [ \Delta X(u) = A_0 X(u-m) + X(u-m) A_1 + G(u) ] is analyzed using recursively defined auxiliary matrices (Q_u) and a fundamental matrix function (Z(u)), yielding closed-form solutions for both homogeneous and non-homogeneous cases. These results extend classical representations for commutative systems to the noncommutative setting. Collectively, continuous and discrete analyzes provide a comprehensive framework for understanding delayed linear matrix dynamics, with potential applications in control theory, signal processing, and iterative learning.