Mathematical Methods

4

Quadratic Forms

4.1 Introduction

Second-degree homogeneous expressions are called quadratic forms. They occur in physics and in geometry. In analytical geometry, for instance, a quadratic form has to be transformed into its principal-axes-form so as to determine the nature of the conic section such as parabola, ellipse or hyperbola, etc., if it involves two variables, and of the quadratic surface such as paraboloid, ellipsoid or hyperboloid, etc., if it involves three variables. A quadratic form can be represented by

where X is a column n-vector and A is a symmetric matrix of the coefficients. We study here the method of transformation of ...