Sunday, October 4, 2009

Sylvester's law of inertia

Let A be a real symmetric square matrix of order n. Any non-singular matrix S of the same size transforms A into another symmetric matrix B of order n defined by the rule
A\to B=SAS^T,
and B is said to be equivalent to A. Transformations of this kind exhibit the effect of a change of coordinates on the matrix of a quadratic form on the n-dimensional real vector space.
A symmetric matrix A can always be transformed into an equivalent diagonal matrix with entries 0, 1 and −1 along the diagonal. Sylvester's law of inertia states that the number of diagonal entries of each kind is an invariant of A, i.e. it does not depend on the matrix S used. The number of 0s, denoted n0, is equal to the dimension of the kernel of A, and also the corank of A. The number of 1s, denoted n+, is called the positive index of inertia, the number of −1s, denoted n−, is called the negative index of inertia and their difference the signature of A:
\operatorname{sign}(A)=n_{+}-n_{-}.
(Sometimes the term signature refers to the whole triple (n0, n+, n−) consisting of the corank and the positive and negative indices of inertia of A). These numbers satisfy an obvious relation
n_0+n_{+}+n_{-}=n.\
If the matrix A has the property that every principal upper left k×k minor Δk is non-zero then the negative index of inertia is equal to the number of sign changes in the sequence
\Delta_0=1, \Delta_1, \ldots, \Delta_n=\det A.
The positive and negative indices of inertia of A can also be characterized as the numbers of positive and negative eigenvalues of A, but this characterization is harder to use in practice.

[edit] Law of inertia for quadratic forms

In the context of quadratic forms, a real quadratic form Q in n variables (or on an n-dimensional real vector space) can by a suitable change of basis be brought to the diagonal form
Q(x_1,x_2,\ldots,x_n)=\sum_{i=1}^{n}a_i x_{i}^2
with each ai∈{0,1,−1}. Sylvester's law of inertia states that the number of coefficients of a given sign is an invariant of Q, i.e. does not depend on a particular choice of diagonalizing basis. Expressed geometrically, the law of inertia says that all maximal subspaces on which the restriction of the quadratic form is positive definite (respectively, negative definite) have the same dimension. These dimensions are the positive and negative indices of inertia.