MetrikSignatur

MetrikSignatur, or metric signature, is a concept in real linear algebra and differential geometry that describes the pair (p, q) associated with a real symmetric bilinear form B on a finite-dimensional real vector space V. It records the numbers of positive and negative squares that appear in a diagonalized representation of B and indicates whether the metric is definite or indefinite.

Sylvester’s law of inertia states that for any real symmetric matrix representing B in any basis, the numbers p and q are invariant under congruence transformations. If B is non-degenerate, p + q equals the dimension of V, and B can be diagonalized by a suitable basis to diag(1, ..., 1, -1, ..., -1) with p ones and q minus ones. If B is degenerate, there is a null space of dimension r, and p + q + r = dim V, where r is called the radical or nullity.

Examples illustrate common signatures. The standard Euclidean inner product on R^n has signature (n, 0). The Minkowski metric used in special relativity has a Lorentzian signature, often written as (1, n−1) or (n−1, 1) depending on convention, e.g., diag(-1, 1, 1, 1). In general relativity and other theories, a pseudo-Riemannian metric typically has a signature (p, q) with p and q determined by the model.

In complex vector spaces, the notion of signature is less standard; positivity is usually discussed in terms of definiteness of Hermitian forms. The metric signature is a fundamental invariant for classifying real quadratic forms and pseudo-Riemannian metrics.