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Invariant

An invariant is something that does not change under a set of transformations. The property of being an invariant is invariance. For the laymen, let us just say an invariant is some kind of correspondence between two types of mathematical objects, so that two 'similar' things correspond to one and the same object. Invariants are useful in discriminating complicated objects.

Mathematicians say that a quantity is invariant "under" a transformation; some economists say it is invariant "to" a transformation.

Some examples, taking more complicated objects to numbers:

• The degree of a polynomial, under linear change of variables
• The dimension of a topological object, under homeomorphism
• The number of fixed points of a dynamical system is invariant under many mathematical operations.
• Euclidean distance is invariant under orthogonal transformations and under translations.
• The cross-ratio is invariant under projective transformations.
• The determinant and the trace of a square matrix are invariant under changes of basis.
• The singular values of a matrix are invariant under orthogonal transformations.
• Lebesgue measure is invariant under translations.
• The variance of a probability distribution is invariant under translations of the real line; hence the variance of a random variable is unchanged by the addition of a constant to it.

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In physics, invariants are usually quantities conserved (unchanged) by the symmetries of the physical system. (See Noether's theorem.) The correspondance between symmetries and conserved quantities is apparent through conservation laws. Much work has been done, esp. in quantum physics, to relate every conserved quantity to some symmetry. One such quantity that still defies all such attempts is mass. Some other examples include:

• Acceleration is invariant under the Galilean transformations.
• The speed of light invariant under the Lorentz transformations of special relativity.
• time invariance makes energy a conserved quantity, hence an invariant.
• translational invariance implies that momentum is conserved.
• rotational invariance implies that angular momentum is conserved.

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In computer science, optimising compilers and the methodology of design by contract pay close attention to invariant quantities in computer programs, where the set of transformations involved is the execution of the steps of the computer program.

A loop invariant is a constraint that should be satisfied by every iteration of a loop.

A class invariant is a constraint that should be satisfied by every method of a class.

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In music using the twelve tone technique invariance describes the portions of rows which have been so designed that they remain invariant under the allowable transformations (inversion, retrograde, retrograde-inversion, multiplication). George Perle describes their use as "pivots" or non-tonal ways of emphasizing certain pitches. Invariant rows are also combinatorial.

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See also

• fixed point (mathematics)
• covariant, contravariant
• principle of relativity
• invariant theory