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.
More generally for any number of variables it can be written as:
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.
