(1) (2)

Prove that if is unitary then is orgthogonal:

As , if is unitary: Then, (because they are the real part) and (because they are the imaginary part).

Therefore: So the statement follows.

Prove that if is orthogonal then is unitary:

If then as shown above, it must be the case that and , which later leads to .

So the statement follows.

Therefore, is orthogonal if and only if is unitary.

(3) Thanks to this:

Let the matrix be , we have: So the eigenvalues of the matrix are , , and .

(4)

Let be the -th element of the matrix , be the -th element of .

According to Euler’s formula: , let , so: So the diagonal elements of equals to . Now we prove that other elements (with ) of equals :   We observe that:   For , . For , there exists a with .

We also observe that:   Working… A note here: What if is odd? Then will not be an integer number. And what if is even?

(5)  If is unitary then :   We also employ the fact that :   Without loss of generality, it is either the case or . , and suggests that in case or in case .

In case : Remember: . So: Therefore a unitary matrix of size with determinant has a form of: So the statement follows.

(6)

From , , , we can derive the general form of a unitary matrix of size : 