where is positive-definite on and negative-definite on that is, for alland for allEquality holds if and only if . The subspace is unique but the subspaces and are not unique. Nevertheless, Sylvester's law of inertia states that the dimensions of  and  are uniquely determined by .

Search through the set of vectors {to find a vector Let be the orthogonal complement of by  Write  If   then or  Continue in this way to arrive at (*). Note that this method does not require the calculation of the eigenvalues/eigenvectors of  

Example 1.

Find the signature of 4 different quadratic forms defined on the tangent space at a point of a 4-dimensional manifold.

First quadratic form.

We see that the quadratic form is positive-definite in all directions; it is a Riemannian metric.

Second quadratic form.

The quadratic form is positive-definite in the 3 directions [ and negative-definite in the 1 direction ; it is a Lorentzian metric.

Third quadratic form.

The quadratic form is positive-definite in the 2 directions [ and negative-definite in the 2 directions [- .

Fourth quadratic form.

The quadratic form is positive-definite in the 2 directions [ and negative-definite in the direction [ and degenerate in the direction [. Here are the dimensions of these spaces.

Example 2.

We calculate the signature of the quadratic forms restricted to some subspaces.

Example 3.

Here we consider quadratic forms which depend upon a parameter.

In this simple example, it is clear that the signature will change depending on the sign of

For more complicated examples, use infolevel to trace the testing performed by the  procedure to see exactly at what point in the algorithm the procedure returns .

Now set the infolevel to 2.

We see that the signature depends on the sign of . 

DifferentialGeometry, Tensor, DGGramSchmidt, SubspaceType