This is one in a series of posts working through the exercises in the Quantum Country online introduction to quantum computing and related topics. The exercises in the original document are not numbered; I have added my own numbers for convenience in referring to them.
Exercise 5. Show that the matrix is unitary.
Answer: A matrix is unitary if where is the adjoint matrix to (produced by taking the transpose of and then replacing all values by their complex conjugates) and is the identity matrix.
Since is symmetric we have and since the values of (and thus of ) are all real we have . We thus have
Since the matrix is unitary.