# Todai Entrance Exam: Math 2019 – Problem 1

**(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 :