Definite quadratic form

In mathematics, a definite quadratic form is a special math expression. It uses squares of numbers from a real space. This form always gives the same kind of result—either always positive or always negative—for every non-zero mix of numbers.

When it always gives positive results, it is called positive-definite. When it always gives negative results, it is called negative-definite.

There is also a semidefinite (or semi-definite) quadratic form. This type never goes below zero or above zero. It can sometimes be zero even when the numbers aren’t all zero.

An indefinite quadratic form is different. It can give both positive and negative results. This kind is known as an isotropic quadratic form.

These ideas work not just with real numbers, but also in any space where numbers can be compared in order, called an ordered field.

Associated symmetric bilinear form

Quadratic forms are closely related to symmetric bilinear forms. Both can be definite, semidefinite, and more, based on their properties.

A quadratic form Q and its related symmetric bilinear form B follow these rules:

• Q(x) equals B(x, x)
• B(x, y) equals B(y, x), and this is also half of [Q(x + y) minus Q(x) minus Q(y)]

These rules show how quadratic forms and symmetric bilinear forms are connected.

Examples

Let's look at some examples of quadratic forms. Imagine a space with two directions, like up-down and left-right. We can create a special rule that tells us how "big" a point is in this space by using numbers for each direction.

If both numbers are positive, our rule will always give a positive result for any point that isn't exactly at the center (where both directions are zero). This is called a positive-definite form.

If one number is positive and the other is zero, the result will either be zero or positive. This is called a positive semidefinite form.

We can also make more complex rules that mix the two directions together. Whether these rules are always positive, always negative, or sometimes positive and sometimes negative depends on the specific numbers we choose. This helps us understand different kinds of quadratic forms.

Optimization

Definite quadratic forms help us solve optimization problems. When we add linear terms to the quadratic form, we get an expression like xᵀAx + bᵀx. To find the highest or lowest value, we set the derivative to zero, which gives us 2Ax + b = 0. Solving this, we find x = −½A⁻¹b, if A can be reversed.

If the quadratic form is positive-definite, this solution gives the lowest value. If it is negative-definite, the solution gives the highest value. This idea is important in multiple regression, where we find the best fit by making the sum of squared differences in the data as small as possible.