Multivariate polynomial where each term has degree 2, e.g.:is a quadratic form because each term has degree 2:but e.g.:is not because the term has degree 3.
There is a 1-to-1 relationship between quadratic forms and symmetric bilinear forms. In matrix representation, this can be written as:where contains each of the variabes of the form, e.g. for 2 variables:
Strictly speaking, the associated bilinear form would not need to be a symmetric bilinear form, at least for the real numbers or complex numbers which are commutative. E.g.:But that same matrix could also be written in symmetric form as:so why not I guess, its simpler/more restricted.
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A quadratic form on a vector space over a field is a function satisfying whose polar bilinear form of a quadratic form is a bilinear form. In finite dimension, it is represented in coordinates by a homogeneous degree-two polynomial, or by with a suitable matrix .
When the characteristic is not two, one may choose to be a symmetric matrix and recover it from . In characteristic two, the symmetric-matrix restriction would lose examples: has an upper-triangular representing matrix , while for a symmetric matrix has no cross term. Its polar form is , an alternating bilinear form. The distinction matters when studying quadratic forms over finite fields.[ref]
A quadratic form is a specific type of polynomial expression that involves variables raised to the second power, usually in the context of multiple variables.