Schur’s second lemma describes the restrictions on a matrix that commutes with elements of two distinct irreducible representations, which may have different dimensions.
Consider an arbitrary matrix and two irreducible representations of a group, of dimension and of dimension , such that
where .
Taking the conjugate transpose of eq10 and using the matrix identity , we have . As every element of a representation of a group can be expressed as a unitary matrix via a similarity transformation without any loss of generality, and as ,
Since the inverse property of a group states that and , we can express eq10 as
Multiplying eq11 on the left by and using eq12, we have and therefore , which implies that commutes with all elements of an irreducible representation of . With reference to Schur’s first lemma,
where is a constant and is the identity matrix.
If we multiply eq11 on the right by and repeat the steps above, we have
Let’s consider the following cases for eq13:
Case 1:
Let the th entry of be . If , we can rewrite eq13 in terms of matrix entries: . If , we have , which implies that is the zero matrix because for all .
Combining eq13 and eq14, we have or . This implies that exists if . We can therefore rewrite eq10 as , which is a similarity transformation if .
Case 2:
If , the arbitrary matrix (denoted by ) is an matrix with reference to eq10. Suppose ; we have:
If we enlarge to form an matrix with the additional elements equal to zero, we have
Due to the zeroes, . Taking the determinants,
Using the determinant identities , and , we have
Since one of the columns of is zero, . So, , which implies that must be a zero matrix according to the results of case 1.
Finally, we can summarise Schur’s second lemma as follows:
Given an arbitrary matrix and two irreducible representations, of dimension and of dimension , where , then

 if , either or the representations are related by a similarity transformation, i.e. equivalent representations.
 If , .