An invariant is something that does not change under a set of transformations. The property of being an invariant is called invariance.

Examples include:

• Euclidean distance is invariant under orthogonal transformations.
• 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.
• Acceleration is invariant under the Galilean transformations.
• The speed of light invariant under the transformations of special relativity.

An important connection exists between conservation laws and the invariance of physical laws with respect to certain transformations. For instance, time invariance implies that energy is conserved, translational invariance implies that momentum is conserved, and rotational invariance implies that angular momentum is conserved.

In mathematics, a fixed point is a value that is an invariant under a mathematical operation such as the application of a mathematical function.

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.

In music using the twelve tone technique invariance describes the portions of rowss 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.

See also:

• principle of relativity