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

$$Q(x_1, x_2, \ldots, x_n)=a_{ij}x_ix_j,$$ (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

$$Q({\bf x})={\bf x}^{\rm T}{\hbox{\sf A}}{\bf x},$$ (2)

equivalent to

$$Q({\bf x})=({\bf x},{\hbox{\sf A}}{\bf x})$$ (3)

in Inner Product notation. A Binary Quadratic Form 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

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

in the form

$$Q({\bf x})=({\bf x},{\hbox{\sf A}}{\bf x}),$$ (6)

where ${\hbox{\sf A}}=a_{ii}$ is a Symmetric Matrix given by

$$a_{ij}=\cases{ \alpha_{ii} & $i=j$\cr {\textstyle{1\over 2}}(\alpha_{ij}+\alpha_{ji}) & $i\not=j$.\cr}$$ (7)

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

$$Q({\bf x})=\lambda_1{x_1}^2+\lambda_2{x_2}^2+\ldots+\lambda_n{x_n}^2$$ (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

References

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.