Quadratic Forms, An Introduction

Introduction

A quadratic equation has a variable x that is squared, and possibly a linear term, and possibly a constant term. (Quad often means 4 in English, as in quadrilateral or quadruplets, but in latin, quad meant square, and the word quadratic reflects this early etimology.) In general, a quadratic form has many variables, and each term has degree 2 or less. An example is shown below.

7x² + 13y² - 17z² + 11xy - 5xz + 9x + 4y - 67 = 0

This is a 3 dimensional example; it defines a quadratic surface in 3 space. You are probably more familiar with quadratic forms in two variables. These generate the conic sections in the plane: parabola, ellipse, and hyperbola. We'll get to all this later, but first, let's talk about quadratic forms and matrices.

Let M be a fixed matrix of real numbers and let x be an unspecified vector in Rⁿ. The components of x are indeterminants, i.e. variables. The quadratic form associated with M is x*M*x^T, using standard matrix multiplication. Expand this all out, and the coefficient on the squared term x_i² is M_i,i, while the coefficient on the mixed term x_ix_j is M_i,j+M_j,i.

Note that there are different matrices that produce the same quadratic form. We could subtract 1 from M_i,j and add 1 to M_j,i, and the result would be the same. By convention, M is assumed to be symmetric.

Conversely, every quadratic form comes from a symmetric matrix M. Take the coefficient on x_i² and place it in M_i,i. Take half the coefficient of x_ix_j and place this in M_i,j and M_j,i.

If M is diagonal, the result is a diagonal form. There are no mixed terms, no linear terms, no constant terms; just a series of squared terms. If we can characterize the diagonal forms, we're halfway home. This is because a rigid rotation turns the quadratic form into a diagonal form. More on this later.