Law of inertia

for quadratic forms

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

$$ \sum _ { i,j=1 } ^ { s } a _ {ij} x _ {i} x _ {j} $$

with real coefficients to a sum of squares

$$ \sum _ { i=1 } ^ { s } b _ {i} y _ {i} ^ {2} $$

by a linear change of variables

$$ (x _ {1} \dots x _ {s} ) = (y _ {1} \dots y _ {s} ) Q , $$

where $Q$ is a non-singular matrix with real coefficients, the number $p$ (respectively, $n$) of indices $ i $ for which $ b _ {i} > 0 $ (or $b_{i} < 0$) 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 $ E $ be a finite-dimensional vector space over an ordered field $ k $, endowed with a non-degenerate symmetric bilinear form $ f $. Then there exists an integer $ p \geq 0 $ such that for any orthogonal basis $ e _ {1} \dots e _ {s} $ in $ E $ with respect to $ f $ there exist among the $ s $ elements

$$ f (e _ {i} , e _ {i} ) ,\ \ i = 1 \dots s , $$

exactly $ p $ positive and exactly $ n = s - p $ negative ones. The pair $ (p , n) $ is called the signature of $ f $, and the number $ n $ its index of inertia. Two equivalent forms have the same signature. If $ k $ is a Euclidean field, equality of signatures is a sufficient condition for the equivalence of bilinear forms. If the index of inertia $ n = 0 $, the form is called positive definite, and for $ p = 0 $, negative definite. These cases are characterized by the property that $f(x, x) > 0$ (respectively, $ f (x , x ) < 0 $) for any non-zero $ x \in E $. It follows from the law of inertia that $ E $ is an orthogonal direct sum (with respect to $f$) of subspaces,

$$ E = E _ {+} \oplus E _ {-} , $$

such that the restriction of $ f $ to $ E _ {+} $ is positive definite while the restriction of $ f $ to $ E _ {-} $ is negative definite and

$$ \mathop{\rm dim} E _ {+} = p ,\ \mathop{\rm dim} E _ {-} = n $$

(so that the dimensions of $ E _ {+} $ and $ E _ {-} $ do not depend on the decomposition).

Sometimes the signature of $ f $ is taken to be the difference

$$ \sigma (f) = p - n . $$

If two forms $ f $ and $ g $ determine the same element of the Witt ring $ W(k) $ of the field $ k $, then $ \sigma (f) = \sigma (g) $. Furthermore, $ \sigma (f _ {1} \oplus f _ {2} ) = \sigma (f _ {1} ) + \sigma (f _ {2} ) $ and $ \sigma (f _ {1} \otimes f _ {2} ) = \sigma (f _ {1} ) \sigma (f _ {2} ) $ for any non-degenerate forms $ f _ {1} $ and $ f _ {2} $, and $ \sigma (\langle 1 \rangle ) = 1 $, so that the mapping $ f \rightarrow \sigma (f) $ defines a homomorphism from $ W (k) $ into the ring of integers $ \mathbf Z $. If $ k $ 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 $k$, over a quadratic extension of $k$ or over the skew-field of quaternions over $k$ (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) MR2333539 MR2327161 MR2325344 MR2284892 MR2272929 MR0928386 MR0896478 MR0782297 MR0782296 MR0722608 MR0682756 MR0643362 MR0647314 MR0610795 MR0583191 MR0354207 MR0360549 MR0237342 MR0205211 MR0205210
[2]	S. Lang, "Algebra" , Addison-Wesley (1974) MR0783636 Zbl 0712.00001
[3]	E. Artin, "Geometric algebra" , Interscience (1957) MR1529733 MR0082463 Zbl 0077.02101
[4]	J. Milnor, D. Husemoller, "Symmetric bilinear forms" , Springer (1973) MR0506372 Zbl 0292.10016

Comments

The name index is also used for $ \min ( p, n) $.