Invariant (mathematics)

In mathematics we mean by an invariant of an associated object to a size that does not change with each matching a class of modifications of the object. Invariants are an important tool in classification problems: objects with different invariants are essentially different; the converse, that is, objects with the same invariants are substantially identical is true, then one speaks of a complete set of invariants or of separating invariants.

Introductory Example

The considered objects are pairs of real numbers, allowed modifications are to add two numbers to the same arbitrarily chosen number:

An invariant in this case is the difference of two numbers:

One interpretation of this example would be: and the starting and end point of a rod, measured from a fixed point in the extension of the rod. The modifications correspond to a displacement of the rod, the invariant is the length of the rod.

In this example, this is already sufficient for a complete invariant classification: Two pairs of numbers and then go right out apart, that is, there is a so that

If the lengths match:

( Proof: Set, then )

Other examples

• The dimension of a vector space is an isomorphism - invariant, that is, and are isomorphic vector spaces, so tune their dimensions match. It is the converse: Two vector spaces of the same dimension ( taken as cardinal number ) on a common base are all isomorphic.
• The determinant of a matrix is ​​a Ähnlichkeitsinvariante, that is, and are two matrices, for which there is an invertible matrix, so that is true, so you have the same determinant. Here, the converse is not true, for example, each rotation has determinant 1
• The Frobenius normal form or Invariantenteiler the characteristic matrix, the unit matrix of the same dimension as A, however, is even a separating invariant of the similarity operation, i.e., two matrices if and similar to each other if they have the same Frobenius normal shape have.
• Betti numbers and Euler characteristic are topological invariants, that is invariant under homeomorphisms.

Invariant under operations

In group operations you also speaks of invariants: Is a set of points with a action of the group, so hot the points that remain invariant,

Fixed points or invariant points.

More generally, each track is produced by a point by the group operation,

Invariant under the group operation.

Advanced Topics

In theoretical physics is the Noether theorem establishes a link between symmetries and invariants of the action of the time evolution. This is called in physics conserved quantities (examples: energy, momentum, angular momentum ). " Relativistic invariance ", ie invariance under Lorentz transformations have many (by postulate: all ) physical theories, including most prominently the Maxwell's electrodynamics and of course the theories of relativity by Albert Einstein. In contrast to mathematics but is ultimately not axiomatic behind it, but a few particularly meaningful experiments such as the Michelson -Morley experiment of the constancy of the speed of light.