A subset S of the domain U of a mapping T is an invariant set under the mapping when Note that the elements of S are not fixed, but rather the set S is fixed in the power set of U. For example, a circle is an invariant subset of the plane under a rotation about the circle’s center. Further, a conical surface is invariant as a set under a homothety of space.
An invariant set of an operation T is also said to be stable under T. For example, the normal subgroups that are so important in group theory are those subgroups that are stable under the inner automorphisms of the ambient group. Other examples occur in linear algebra. Suppose a linear transformation T has an eigenvector v. Then the line through 0 and v is an invariant set under T. The eigenvectors span an invariant subspace which is stable under T.
Read more about this topic: Invariant (mathematics)
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