Quadratic Form

A quadratic form involving real variables , , ..., associated with the real matrix is given by

(1)

where Einstein summation has been used. Letting be a vector made up of , ..., and the transpose, then

(2)

equivalent to

(3)

in inner product notation. A binary quadratic form is a quadratic form in two variables. For a symmetric matrix , it has the form

(4)

It is always possible to express an arbitrary real quadratic form

(5)

with Einstein summation in the form

(6)

where is a symmetric matrix given by

(7)

Any real quadratic form in variables may be reduced to the diagonal form

(8)

with by a suitable orthogonal transformation. Also, two real quadratic forms are equivalent under the group of invertible linear transformations iff they have the same quadratic form rank and quadratic form signature.

More generally, on a vector space over a field , a quadratic form is a function satisfying for all and , such that is a bilinear form in and (Conrad, pp. 34-35 and 48-49). When has field characteristic different from 2, dividing this bilinear form by 2 gives the unique bilinear form satisfying and .

A norm-Euclidean lattice uses the field norm of a quadratic form to measure approximation by integral vectors.