Every group G acts on itself by inner automorphisms: the map associated with an element g is h ↦ ghg −1. If G is a Lie group, the differential of each inner automorphism determines a linear transformation on the tangent space to G at the identity element, because the identity is fixed by any inner automorphism. This gives rise to a linear representation of the group G in the Lie algebra G of G, the adjoint representation of G. If G is a matrix group, the Lie algebra G, too, is realized by matrices, and the adjoint representation τ g maps x ∈ G to gxg −1.