A homogeneous polynomial of degree 2 in is called a quadratic form (over ) in indeterminates. In general, a quadratic form (without specifying ) over a ring is a quadratic form in some polynomial ring over .
For example, in , is a quadratic form, while and are not.
In general, a quadratic form in -indeterminates looks like
Letting , and the matrix, then we can rewrite as
For example, the quadratic form can be rewritten as
Again, in the example of , over it can be written as
However, it is not possible to represent over by a symmetric matrix.
Evaluating a Quadratic Form
It is not hard to see that, given a quadratic form in indeterminates, setting one of its indeterminates to gives us another quadratic form, in indeterminates. This is an informal way of saying the following:
In particular, if we take , and with . Then the evaluation homomorphism at for any quadratic form is called the evaluation of at , and we write it , or simply (since is uniquely determined by ). In this way, a quadratic form can be realized as a quadratic map, as follows:
Let be a qudratic form. Take the direct sum of copies of and call this . Define a map by . Then is a quadratic map.
Conversely, if is invertible in (so that is clear), then given a quadratic map , one can find a corresponding quadratic form such that , by setting
where and are coordinate vectors whose coordinates are all except at positions and respectively, where the coordinates are . Then defined by , where is the desired quadratic form.
Equivalence of Quadratic Forms
From the above discussion, we shall identify a quadratic form as a quadratic map.
Two quadratic forms and are said to be if there is an invertible matrix such that , for all . The definition of equivalent quadratic forms is well-defined and it is not hard to see that this equivalence is an equivalence relation.
In fact, if and are matrices corresponding to (see the definition section) the two equivalent quadratic forms and above, then .
For example, the quadratic form is equivalent to over any ring where is invertible, with .
In the case where is the field of real numbers (or any formally real field), we say that a quadratic form is positive definite, negative definite, or positive semidefinite according to whether its corresponding matrix is positive definite, negative definite, or positive semidefinite. The definiteness of a quadratic form is preserved under the equivalence relation on quadratic forms.
Sums of Quadratic Forms
If and are two quadratic forms in and indeterminates. We can define a quadratic form in indeterminates in terms of and , called the sum of and , as follows:
write and , with and . Then
where , and is the direct sum of matrices and .
Expressed in terms of and , we write . For example, if and , then
- 1 T. Y. Lam, Introduction to Quadratic Forms over Fields, American Mathematical Society (2004)
|Date of creation||2013-03-22 12:19:22|
|Last modified on||2013-03-22 12:19:22|
|Last modified by||rm50 (10146)|
|Defines||equivalent quadratic forms|
|Defines||sum of quadratic forms|
|Defines||evaluation of a quadratic form|