<![CDATA[This article examines the application of the spectral theorem for symmetric matrices in the modeling of linear dynamical systems. The study covers proofs of core properties of symmetrical matrices (real eigenvalues, orthogonality of eigenvectors associated with distinct eigenvalues, and orthogonal diagonalization), and applies these theoretical results to solve low-dimensional linear dynamical systems commonly found in vibration models and simple mechanical systems. In addition to theoretical proofs, the paper includes computational examples (2×2 and 3×3 systems), stability analysis via matrix spectra, and pedagogical implications for teaching linear algebra at the university level. The computational results show that the spectral theorem effectively predicts system behavior through eigenvalue signs: positive eigenvalues produce exponential divergence, while negative eigenvalues produce convergence to equilibrium. Visualization confirms that trajectories align with the dominant eigenvector direction, demonstrating the geometric significance of spectral decomposition. This study also highlights the pedagogical impact of integrating spectral analysis with computational tools, showing that visualization-supported explanations improve students’ conceptual understanding of eigenvalues, eigenvectors, and system stability. The approach supports deeper learning in university-level mathematics education.]]>
Copyrights © 2026
