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An invariant in mathematics 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.

Topics: Invariant (mathematics), Invariant Mass, Invariant Theory...

InvariantInvariant may have meanings invariant (computer science), such as a combination of variables not altered in a loop invariant (mathematics), something unaltered by a transformation invariant (music) invariant (physics) conserved by system symmetry.	Invariant massIn particle physics, the mathematical combination of a particle's energy and its momentum to give a value for the mass of the particle at rest. The invariant mass is the same for all frames of reference (see Special Relativity). The invariant mass of a sy	Invariant theoryIn mathematics, invariant theory refers to the study of invariant algebraic forms (equivalently, symmetric tensors) for the action of linear transformations. This was a major field of study in the latter part of the nineteenth century, when it appeared th
Invariant subspaceLinear algebra Functional analysis Operator theory In mathematics, an invariant subspace of a linear mapping over some vector space V is a subspace W of V such that : , literally: T ''W is contained in W''. An invariant subspace of T is said to be T invar	Invariant (mathematics)An invariant in mathematics 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	Invariant (physics)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 d
Invariant (computer science)	Invariant speed

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Topics: Invariant (mathematics), Invariant Mass, Invariant Theory...