Sylvester's Inertia Theorem

This lecture covers Sylvester's Inertia Theorem, which states that for a real symmetric matrix A, the number of positive, negative, and zero eigenvalues is equal to the number of positive, negative, and zero diagonal entries in the diagonalized form of A. The theorem is proven using the concept of Sylvester's Law of Inertia and the signature of a symmetric matrix. The lecture also discusses the construction of Sylvester's Orthogonal Basis and the properties of the Gram matrix of a bilinear form. The importance of Sylvester's Inertia Theorem in understanding the definiteness of symmetric matrices is highlighted.