ABSTRACT
In this chapter we introduce the notion of a quadratic form. Quadratic forms are prevalent in linear algebra and has many applications. In Section 7.1 we introduce the notion of a quadratic form and the associated definitions of positive and negative definite and semi-definite and indefinite. In Section 7.2 we derive the first test for determining if a quadratic form is positive or negative, definite or semi-definite, or indefinite called the Principal Minor Criterion. In Section 7.3 we derive the second test for determining if a quadratic form is positive or negative, definite or semi-definite, or indefinite called the Eigenvalue Criterion. In Section 7.4 we apply the criteria developed in Sections 7.2 & Section 7.3 to analyze critical points to determine if they are extrema for a multivariate function. In Section 7.5 we generalize the notion of a quadratic form.
