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

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 \ Ẋ () = A₀ X (-σ) + X (-σ) A₁ + G (), 0, \ where \ (A₀, A₁ R^d d\) satisfy \ (A₀ A₁ A₁ A₀\), \ (G () \) is a prescribed matrix function, and \ (σ>0\) denotes the delay. We derive explicit representations of the solution for initial data \ (X () = Ψ () \), \ (-σ, 0\), highlighting the structural impact of noncommutativity. For the discrete-time analogue, the system \ ΔX (u) = A₀ X (u-m) + X (u-m) A₁ + G (u) \ is analyzed using recursively defined auxiliary matrices \ (Qᵤ\) 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.