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Quadratic Form

A quadratic form involving n real variables x_1, x_2, ..., x_n associated with the n×n matrix A=a_(ij) is given by

Q(x_1,x_2,...,x_n)=a_(ij)x_ix_j,
(1)

where Einstein summation has been used. Letting x be a vector made up of x_1, ..., x_n and x^(T) the transpose, then

Q(x)=x^(T)Ax,
(2)

equivalent to

Q(x)=<x,Ax>
(3)

in inner product notation. A binary quadratic form is a quadratic form in two variables and has the form

Q(x,y)=a_(11)x^2+2a_(12)xy+a_(22)y^2.
(4)

It is always possible to express an arbitrary quadratic form

Q(x)=alpha_(ij)x_ix_j
(5)

in the form

Q(x)=(x,Ax),
(6)

where A=a_(ii) is a symmetric matrix given by

a_(ij)={alpha_(ii) i=j; 1/2(alpha_(ij)+alpha_(ji)) i!=j.
(7)

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

Q(x)=lambda_1x_1^2+lambda_2x_2^2+...+lambda_nx_n^2
(8)

with lambda_1>=lambda_2>=...>=lambda_n 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.

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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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