# Quadratic forms with Applications

Thesis

The scope of Quadratic Form Theory is historically wide although it usually appears almost as an afterthought when needed to solve a variety of problems such as the classification of Hessian matrices in finite dimensional Calculus [1], [2], [3], the finding of invariants that fully describe the equivalence class of a given form in Algebraic Geometry and Number Theory [4], the use of Rayleigh-Ristz methods for finding eigenvalues of real symmetric matrices in Linear Algebra [5], [6], the second order optimality conditions in Optimization Theory [1], [2], [3], the Sturm comparison criteria and the Sturm-Liouville Boundary Value Problems in Differential Equations [5], the kinetic energy or the Hamiltonian in Mechanics [8], etc... In Advanced Mathematics, Quadratic Forms occupy a central place in various branches including Number Theory, Algebra, Group Theory (orthogonal groups) [7], [4], [29], Calculus of Variations and Optimal Control Theory (the second variational problem)[9], Operator Theory [8], Differential Geometry (Riemannian metrics and fundamental forms) [11], Morse Theory (Morse lemma) [12], [13], Differential Topology (intersection forms of four-manifolds), and Lie Theory (the Killing form) [14], [15]. In this dissertation, our aim is to review the Theory of Quadratic Forms on Euclidean and Hermitian spaces, to give an idea of its generalization to Hilbert spaces and to mention some common applications including the Linear Regression, the Mean Square Approximation, the Rayleigh-Ristz method and the Lax-Milgram Theorem (the bounded as well as the unbounded cases).