Schur's Lemma - Matrix Form

Matrix Form

Let G be a complex matrix group. This means that G is a set of square matrices of a given order n with complex entries and G is closed under matrix multiplication and inversion. Further, suppose that G is irreducible: there is no subspace V other than 0 and the whole space which is invariant under the action of G. In other words,

Schur's lemma, in the special case of a single representation, says the following. If A is a complex matrix of order n that commutes with all matrices from G then A is a scalar matrix. If G is not irreducible, then this is not true. For example, if one takes the subgroup D of diagonal matrices inside of GL(n,C), then the center of D is D, which contains non scalar matrices. As a simple corollary, every complex irreducible representation of Abelian groups is one-dimensional.

See also Schur complement.