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]).