Linear Algebra with Applications, 10th Edition

6.6 Quadratic Forms

By this time, the reader should be well aware of the important role that matrices play in the study of linear equations. In this section, we will see that matrices also play an important role in the study of quadratic equations. With each quadratic equation, we can associate a vector function $f\left(\mathbf{\text{x}}\right)={\mathbf{\text{x}}}^{T}A\mathbf{\text{x}}$. Such a vector function is called a quadratic form. Quadratic forms arise in a wide variety of applied problems. They are particularly important in the study of optimization theory.

Definition

A quadratic equation in two variables x and y is an equation of the form

$$a{x}^2+2bxy+c{y}^2+dx+cy+f=0$$ (1)

Equation (1) may be rewritten in the form

$$\left[\begin{array}{cc}x& y\end{array}\right]\left[\begin{array}{cc}a& b\\ b& c\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{cc}d& e\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]+f=0$$ (2)

Let

$$\begin{array}{ccc}\mathbf{\text{x}}=\left[\begin{array}{c}x\\ y\end{array}\right]& \text{and}& A=\left[\begin{array}{cc}a& b\\ b& c\end{array}\right]\end{array}$$

The term

$${\mathbf{\text{x}}}^{T}A\mathbf{\text{x}}=a{x}^2+2bxy+c{y}^2$$

is ...