Metric signature

The signature (also inertia index or index ) is an object of mathematics, which is considered especially in linear algebra but also in different areas of differential geometry. Exactly, it is a triple of numbers, which is an invariant of a symmetric bilinear form. This number triples in particular is thus independent of the basis choice, with respect to the bilinear form is displayed. Fundamental to the definition of the signature is the inertia of Sylvester's theorem, named after the mathematician James Joseph Sylvester. Therefore, the signature is also sometimes called Sylvester signature.

Definition

Be a finite dimensional real vector space and a symmetric bilinear form with the matrix representation

This matrix has the entries on the main diagonal, and all other coefficients.

By now, the number of entries, referred to as the number of messages and the number of entries. Then called the triple

Index of inertia or ( Sylvester ) signature. Since, according to the inertia set of New Year's Eve every symmetric bilinear form has a diagonal matrix such as matrix representation, the signature for all symmetric bilinear forms is well-defined.

Standing on the main diagonal of the matrix representation no zero entries, (that is the symmetric bilinear non-degenerate ), then the coefficient is also sometimes omitted, and it is called the tuple

The signature of. It is sometimes also

Called a signature ( in particular, if there is no degeneracy exists ). Sometimes also called index.

The term of the signature is also used for symmetric matrices. He then referred to the signature of the symmetric bilinear form defined by for.

Algorithm for determining the signature

To compute the signature of a symmetric bilinear form, it is not necessary to calculate the change of basis matrix of the representation. After any representation matrix (not necessarily in diagonal form ) was determined symmetric bilinear form, it can also be regarded as a representation matrix of an endomorphism. From this matrix one can then determine the eigenvalues. We then denote by the number of positive eigenvalues ​​, with the number of negative entries, and with the multiplicity of the eigenvalue, then corresponds to

The signature of.

Example

Let be a symmetric bilinear form. Thus, the matrix representing the canonical basis has the form

Summarizing this matrix as a self-adjoint endomorphism of the meantime, so you know, due to the Spectral Theorem that there is an orthonormal basis of eigenvectors, so has diagonal form. Multiplying each eigenvector still with, the corresponding eigenvalue, and then performs the basic transformation, we obtain a diagonal matrix with entries 1 and -1 on the diagonal. Here you can directly read the signature. In our specific example, are the eigenvalues ​​and orthonormal eigenvectors. Multiplying this base yet as described above with, we obtain the transformation matrix

And the basis transformation looks like this:

So the matrix of the associated bilinear form has the signature. In this example, you have to consider that bilinear forms have no eigenvalues ​​and that the path of the eigenvalues ​​is just a trick to calculate with.

The above diagonal form could also be with the Gauss algorithm calculated by transformations always be equally applied to rows and columns.

An important example from physics is the Minkowski metric of special relativity. This is a symmetric bilinear form with signature ( 1,3,0 ) or simply (1.3 ). is the cone of light, the time-like vectors, the space-like vectors. Change from a choice of (, ) to another means a boost.

Special case

Where is a symmetric non- singular matrix. Then the signature is given by:

Herein, the first principal minor of. The other two variables arise in calculating the determinants of other minors, with only the sign is important. is the number of constant sign of by and the number of sign changes from to.

The signature in differential geometry

Signature of a pseudo - Riemannian manifold

In the differential geometry is generalized symmetric bilinear forms on differentiable manifolds in the form of symmetric covariant tensor fields smoother second stage. Such a tensor field then acts in each dot on each tangent space as a bilinear form. If the signature of the respective bilinear same at each point of the manifold and these are non-degenerate, then one speaks of a pseudo - Riemannian metric and identifies a manifold that is provided with such a metric, pseudo - Riemannian manifold. Such manifolds are object of investigation of the pseudo - Riemannian geometry and play an important role in physics.

Signature of a manifold

In the global analysis, a portion of the differential geometry, the signature is considered a manifold. To define the signature of such a " curved space ", a special bilinear form is chosen and determined that their signature is the signature of the manifold. The signature set of Hirzebruch is a central statement in this context. It sets the signature that is an invariant of the bilinear form, with an invariant of the manifold in conjunction.

Let be a compact, orientable smooth manifold whose dimension is divisible by. It also referred to the de Rham cohomology of. Consider the bilinear form defined by

Is defined. This is symmetric and non-degenerate due to the Poincaré duality, that is. Then the signature of the manifold is defined as the signature of bilinear, i.e.