تحليل طيفي لاستقرار حلول المعادلات الخطية باستخدام القيم الذاتية (Comprehensive Journal of Science, 2026)
This study investigates the stability of solutions in linear time-dependent systems through a spectral approach based on the eigenvalues of the system matrix. A mathematically rigorous framework is formulated for both continuous and discrete linear models, establishing the connection between the spectral properties of the matrix and the fulfillment of stability conditions using criteria such as the spectral radius and the distribution of eigenvalues in the complex plane.The methodology involves the analysis of representative cases of matrices with varying structural characteristics, including diagonal, non-symmetric, non-diagonalizable, and algebraically repeated spectra. The results demonstrate that the classical spectral conditions (such as negative real parts of eigenvalues for continuous systems or location within the unit circle for discrete systems) are reliable indicators of stability, provided that algebraic and geometric multiplicities are appropriately considered.The findings confirm that spectral analysis offers a precise and efficient tool for assessing stability without the need for direct time-domain simulation. The study lays the groundwork for extending this approach to more complex systems, including nonlinear or time-varying models.
Keywords: Spectral analysis, Eigenvalues, Solution stability, Linear equations, Time-dependent systems, Spectral radius, Matrix analysis