Definite quadratic form

In mathematics, a definite quadratic form is a special kind of mathematical expression that 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 combination of numbers in that space. When it always gives positive results, it is called positive-definite, and when it always gives negative results, it is called negative-definite.

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

An indefinite quadratic form is different because 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. They both can be described as definite, semidefinite, and so on, 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 relationships help show how quadratic forms and symmetric bilinear forms connect to each other.

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 are useful in solving optimization problems. When we add linear terms to the quadratic form, we get an expression like xᵀAx + bᵀx. To find the maximum or minimum value, we set the derivative to zero, which gives us 2Ax + b = 0. Solving this, we find x = −½A⁻¹b, assuming A is invertible.

If the quadratic form is positive-definite, this solution gives a minimum value. If it is negative-definite, the solution gives a maximum value. This idea is important in multiple regression, where we find the best fit by minimizing the sum of squared differences in the data.