Invariant (mathematics) - More Advanced Examples

More Advanced Examples

Some more complicated examples:

• The real part and the absolute value of a complex number are invariant under complex conjugation.
• The degree of a polynomial is invariant under linear change of variables.
• The dimension and homology groups of a topological object are invariant under homeomorphism.
• The number of fixed points of a dynamical system is invariant under many mathematical operations.
• Euclidean distance is invariant under orthogonal transformations.
• Euclidean area is invariant under a linear map with determinant 1 (see Equi-areal maps).
• The cross-ratio is invariant under projective transformations.
• The determinant, trace, and eigenvectors and eigenvalues of a square matrix are invariant under changes of basis. In a word, the spectrum of a matrix is invariant to the change of basis.
• Invariants of tensors.
• 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.
• The fixed points of a transformation are the elements in the domain invariant under the transformation. They may, depending on the application, be called symmetric with respect to that transformation. For example, objects with translational symmetry are invariant under certain translations.
• The integral of the Gaussian curvature K of a 2-dimensional Riemannian manifold (M,g) is invariant under changes of the Riemannian metric g. This is the Gauss-Bonnet Theorem.