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 .

See also Disconnected Form ,

Indefinite Quadratic Form ,

Inner Product ,

Integer-Matrix
Form ,

Positive Definite Quadratic
Form ,

Positive Semidefinite
Quadratic Form ,

Quadratic ,

Quadratic
Form Rank ,

Quadratic Form Signature ,

Sylvester's Inertia Law ,

Symmetric
Quadratic Form
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References Bayer-Fluckinger, E.; Lewis, D.; and Ranicki, A. (Eds.). Quadratic
Forms and Their Applications: Proceedings of the Conference on Quadratic Forms and
Their Applications, July 5-9, 1999, University College, Dublin. Providence,
RI: Amer. Math. Soc., 2000. Buell, D. A. Binary
Quadratic Forms: Classical Theory and Modern Computations. New York:Springer-Verlag,
1989. Conway, J. H. and Fung, F. Y. The
Sensual (Quadratic) Form. Washington, DC: Math. Assoc. Amer., 1997. Gradshteyn,
I. S. and Ryzhik, I. M. Tables
of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press,
pp. 1104-106, 2000. Kitaoka, Y. Arithmetic
of Quadratic Forms. Cambridge, England: Cambridge University Press, 1999. Lam,
T. Y. The
Algebraic Theory of Quadratic Forms. Reading, MA: W. A. Benjamin,
1973. Weisstein, E. W. "Books about Quadratic Forms."
http://www.ericweisstein.com/encyclopedias/books/QuadraticForms.html . Referenced
on Wolfram|Alpha Quadratic Form
Cite this as:
Weisstein, Eric W. "Quadratic Form." From
MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/QuadraticForm.html

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