Law of inertia

for quadratic forms

The theorem stating that for any way of reducing a quadratic form (cf. also Quadratic forms, reduction of)

with real coefficients to a sum of squares

by a linear change of variables

where is a non-singular matrix with real coefficients, the number (respectively, ) of indices for which (or ) is fixed. In its classical form, the law of inertia was established by J.J. Sylvester. This statement is sometimes called Sylvester's theorem.

In its modern form, the law of inertia is the following statement concerning properties of symmetric bilinear forms over ordered fields. Let be a finite-dimensional vector space over an ordered field , endowed with a non-degenerate symmetric bilinear form . Then there exists an integer such that for any orthogonal basis in with respect to there exist among the elements

exactly positive and exactly negative ones. The pair is called the signature of , and the number its index of inertia. Two equivalent forms have the same signature. If is a Euclidean field, equality of signatures is a sufficient condition for the equivalence of bilinear forms. If the index of inertia , the form is called positive definite, and for , negative definite. These cases are characterized by the property that (respectively, ) for any non-zero . It follows from the law of inertia that is an orthogonal direct sum (with respect to ) of subspaces,

such that the restriction of to is positive definite while the restriction of to is negative definite and

(so that the dimensions of and do not depend on the decomposition).

Sometimes the signature of is taken to be the difference

If two forms and determine the same element of the Witt ring of the field , then . Furthermore, and for any non-degenerate forms and , and , so that the mapping defines a homomorphism from into the ring of integers . If is a Euclidean field, then this homomorphism is an isomorphism.

The law of inertia can be generalized to the case of a Hermitian bilinear form over a maximal ordered field , over a quadratic extension of or over the skew-field of quaternions over (see [1], [4]).

References

[1]	N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French)
[2]	S. Lang, "Algebra" , Addison-Wesley (1974)
[3]	E. Artin, "Geometric algebra" , Interscience (1957)
[4]	J. Milnor, D. Husemoller, "Symmetric bilinear forms" , Springer (1973)

Comments

The name index is also used for .