Quadratic form

A fundamental question is the classification of quadratic form under linear change of variables.

and these two processes are the inverses of each other. As a consequence, over a field of characteristic not equal to 2, the theories of symmetric bilinear forms and of quadratic forms in n variables are essentially the same.

This is the current use of the term; in the past it was sometimes used differently, as detailed below.

Historically there was some confusion and controversy over whether the notion of integral quadratic form should mean:

the quadratic form associated to a symmetric matrix with integer coefficients
a polynomial with integer coefficients (so the associated symmetric matrix may have half-integer coefficients off the diagonal)

This debate was due to the confusion of quadratic forms (represented by polynomials) and symmetric bilinear forms (represented by matrices), and "twos out" is now the accepted convention; "twos in" is instead the theory of integral symmetric bilinear forms (integral symmetric matrices).

Several points of view mean that twos out has been adopted as the standard convention. Those include: