This paper develops a theory for the semi-convergence of dual matrices. We establish necessary and sufficient conditions for semi-convergence and derive an explicit limit formula using the dual group inverse. Extending the classical theory, we introduce dual \(M\) -matrices and property \(C\) , and establish that a dual \(M\) -matrix possesses property \(C\) if and only if its dual group inverse exists. Based on this theoretical framework, we propose and compare dual stationary and dual Chebyshev semi-iterative methods for solving dual linear systems. Numerical experiments on the perturbed Poisson equation confirm that the dual Chebyshev method improves convergence speed and scalability. Finally, we apply these results to dual Markov chains, demonstrating that the semi-convergence limit captures the stationary distribution and its sensitivity to perturbations.
