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Quadratic Form

A quadratic form involving $n$ Real variables $x_1$, $x_2$, ..., $x_n$ associated with the $n\times
n$ Matrix ${\hbox{\sf A}}=a_{ij}$ is given by

Q(x_1, x_2, \ldots, x_n)=a_{ij}x_ix_j,
\end{displaymath} (1)

where Einstein Summation has been used. Letting ${\bf x}$ be a Vector made up of $x_1$, ..., $x_n$ and ${\bf x}^{\rm T}$ the Transpose, then
Q({\bf x})={\bf x}^{\rm T}{\hbox{\sf A}}{\bf x},
\end{displaymath} (2)

equivalent to
Q({\bf x})=({\bf x},{\hbox{\sf A}}{\bf x})
\end{displaymath} (3)

in Inner Product notation. A Binary Quadratic Form has the form
\end{displaymath} (4)

It is always possible to express an arbitrary quadratic form

Q({\bf x})=\alpha_{ij}x_ix_j
\end{displaymath} (5)

in the form
Q({\bf x})=({\bf x},{\hbox{\sf A}}{\bf x}),
\end{displaymath} (6)

where ${\hbox{\sf A}}=a_{ii}$ is a Symmetric Matrix given by
\alpha_{ii} & $i=j$\cr
{\textstyle{1\over 2}}(\alpha_{ij}+\alpha_{ji}) & $i\not=j$.\cr}
\end{displaymath} (7)

Any Real quadratic form in $n$ variables may be reduced to the diagonal form
Q({\bf x})=\lambda_1{x_1}^2+\lambda_2{x_2}^2+\ldots+\lambda_n{x_n}^2
\end{displaymath} (8)

with $\lambda_1\geq \lambda_2\geq \ldots \geq \lambda_n$ by a suitable orthogonal point-transformation. Also, two real quadratic forms are equivalent under the group of linear transformations Iff they have the same Rank and Signature.

See also Disconnected Form, Indefinite Quadratic Form, Inner Product, Integer-Matrix Form, Positive Definite Quadratic Form, Positive Semidefinite Quadratic Form, Rank (Quadratic Form), Signature (Quadratic Form), Sylvester's Inertia Law


Quadratic Forms

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., 1998.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, pp. 1104-106, 1979.

Lam, T. Y. The Algebraic Theory of Quadratic Forms. Reading, MA: W. A. Benjamin, 1973.

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© 1996-9 Eric W. Weisstein