Invariant (mathematics)

In mathematics , an invariant is understood to be a variable associated with an object that does not change with a suitable class of modifications of the object. Invariants are an important aid in classification problems : Objects with different invariants are essentially different; the reverse also applies, i.e. H. if objects with the same invariants are essentially identical, one speaks of a complete set of invariants or of separating invariants .

Introductory example

The objects under consideration are pairs of real numbers ; permitted modifications consist in adding the same arbitrarily chosen number to both numbers: ${\ displaystyle (x, y)}$

${\ displaystyle (x, y) \ quad \ longmapsto \ quad (x ', y') = (x + z, y + z)}$.

In this case, an invariant is the difference between the two numbers: ${\ displaystyle xy}$

${\ displaystyle x'-y '= (x + z) - (y + z) = xy.}$

An interpretation of this example could be: and are the start and end points of a rod, measured from a fixed point in the extension of the rod. The modifications correspond to a shift of the rod by , the invariant is the length of the rod. ${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle z}$

In this example, this one invariant is sufficient for a complete classification: two pairs of numbers and then emerge from each other, that is, there is a such that ${\ displaystyle (x_ {1}, y_ {1})}$${\ displaystyle (x_ {2}, y_ {2})}$${\ displaystyle z}$

${\ displaystyle x_ {1} + z = x_ {2}}$ and ${\ displaystyle y_ {1} + z = y_ {2},}$

if the lengths match:

${\ displaystyle x_ {1} -y_ {1} = x_ {2} -y_ {2}.}$

(Proof: set , then is ) ${\ displaystyle z = x_ {2} -x_ {1}}$${\ displaystyle y_ {1} + z = x_ {2} - (x_ {1} -y_ {1}) = x_ {2} - (x_ {2} -y_ {2}) = y_ {2}.}$

Further examples

• The dimension of a vector space is an isomorphism invariant, i. i.e., if and are isomorphic vector spaces, then their dimensions coincide. The converse also applies: two vector spaces of the same dimension (interpreted as a cardinal number ) over a common base are isomorphic.${\ displaystyle V}$${\ displaystyle W}$
• The determinant of a matrix is a similarity invariant; i.e., and are two matrices for which there is an invertible matrix such that so and have the same determinant. The converse does not apply here, for example every rotation has the determinant 1.${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle S}$${\ displaystyle B = SAS ^ {- 1}}$${\ displaystyle A}$${\ displaystyle B}$
• The Frobenius normal form or the invariant divisors of the characteristic matrix , where the identity matrix is ​​of the same dimension as A, but is even a separating invariant of the similarity operation, i.e. H. two matrices are similar to each other if and only if they have the same Frobenius normal form.${\ displaystyle xI-A}$${\ displaystyle I}$
• Betti numbers and Euler characteristics are topological invariants ; H. invariant under homeomorphisms .

Invariants under operations

In group operations can also speak of invariants: Is a set of points with an operation of the group , so the names of the points that remain invariant, ${\ displaystyle X}$ ${\ displaystyle G}$

${\ displaystyle x \ in X: \, gx = x \ \ mathrm {f {\ ddot {u}} r \ alle} \ g \ in G}$

Fixed points or the invariant points. ${\ displaystyle G}$

More generally, every path through a point is created by the group operation, ${\ displaystyle x \ ,,}$

${\ displaystyle Gx = \ {y \ in X \ mid \, y = gx \ ,, g \ in G \}}$

invariant under the group operation.

Further topics

In theoretical physics , the Noether theorem establishes a connection between symmetries of action and invariants of time evolution. In physics, these are called conserved quantities (examples: energy, momentum, angular momentum). "Relativistic Invariance", d. H. Invariance against Lorentz transformations have many (by postulate: all ) physical theories, including Maxwell's electrodynamics and of course Albert Einstein's theories of relativity . In contrast to mathematics, however, it is ultimately not based on axiomatics , but rather a few particularly meaningful experiments such as the Michelson-Morley experiment on the constancy of the speed of light.