A unitary representation of a group consists of elements that are unitary matrices.

Every representation of a group can be described in terms of unitary matrices. Specifically, matrices of a representation of a group can be expressed as unitary matrices via a common similarity transformation without any loss of generality. The proof is as follows:

Consider a group , where each element is an matrix. Let’s construct a new matrix out of the elements of , where .

###### Question

Proof the matrix identity .

###### Answer

Using the identity mentioned above,

Therefore, is a Hermitian matrix, which can be diagonalised by a unitary matrix , i.e. . Using and the above identity again, we have,

or

where .

The diagonal elements of are and hence, the diagonal elements of are . Moreover, one of the elements of is the identity matrix, with and . So, .

###### Question

Why is ?

###### Answer

implies that is a complex number. The modulus of a complex number is . Hence, .

Let and be

where , and .

Consider a new set of matrices , where . Since is diagonal and is real and positive, we have . Therefore, and

Substitute eq1 in the above equation and changing the dummy index from to ,

###### Question

Show that the set is also a representation of .

###### Answer

If the set is also a representation of , its elements must multiply according to the multiplication table of . Since, , we have . The third equality ensures that the closure property of is satisfied for the set and hence the set . In other words, the elements of multiply according to the multiplication table of .

If , then . Since, , the only possibility is that for . Therefore, the set has the identity element.

To show that each has an inverse, we have

Finally, the associativity property of the group is evident, due to the fact that the set consists of matrices, which are associative.

Since the set is also a representation of , we can express eq2 as:

###### Question

Explain why the 2nd equality in the above equation is valid.

###### Answer

According to the rearrangement theorem, each summand in is a unique element of , which is denoted by . Therefore, the 2nd equality in the equation before the Q&A holds.

Repeating the steps from eq2 for , we have . Therefore, is a unitary matrix.

Since , and , we have

In other words, every element of a representation of can undergo a similarity transformation that results in , which is unitary. This is necessary in proving Schur’s first lemma and Schur’s second lemma.

###### Question

Show that the set is also a representation of .

###### Answer

If the set is also a representation of , its elements must multiply according to the multiplication table of . Since , we have

The third equality ensures that the closure property of is satisfied for the set and hence the set . In other words, the elements of multiply according to the multiplication table of .