“Quadratic Form”的概念、定义、习题解题方法及翻译-数学辞典

单词

Quadratic Form

释义

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 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 realquadratic 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.

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