Quadratic Form

A quadratic form involving real variables , , ..., associated with the 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 and has the form

(4)

It is always possible to express an arbitrary quadratic form

(5)

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 point-transformation. Also, two real quadratic forms are equivalent under the group of linear transformations iff they have the same quadratic form rank and quadratic form signature.

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