The Spectral Theorem - Every Symmetric Matrix Has n Orthogonal Eigenvectors

Learn the fundamental spectral theorem through a detailed mathematical proof demonstrating that every symmetric matrix of order n×n possesses exactly n orthogonal eigenvectors. Explore the theoretical foundations and step-by-step proof methodology that establishes this crucial result in linear algebra, understanding how symmetric matrices can be diagonalized using orthogonal transformations and why this property makes them particularly important in mathematical analysis and applications.