Question: Show that a symmetry operator R must be unitary, that is, its adjoint is equal to its inverse; for a matrix representation this means that

Show that a symmetry operator R must be unitary, that is, its adjoint is equal to its inverse; for a matrix representation this means that its inverse is equal to the complex conjugate of the transposed matrix. There are several steps to showing this.

(a) Show that

(b) Inserting a sum over a complete set of states into

By evaluating this, show that

and therefore R is unitary. Note that this analysis also implies that only one row in any column is nonzero, and vice versa.

|ij | = 1.

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