**Prove that a matrix that is both unitary and upper triangular must be a diagonal matrix.**

In case you don't remember what a unitary matrix is, please refer to wikipedia.

Possibly relevant theorem from the textbook:

Theorem 6.18. Let

be a linear operator on a finite-dimensional inner product space

. Then the following statements are equivalent:

(a)

.

(b)

for all

.

(c) If

is an orthonormal basis for

, then

is an orthonormal basis for

.

(d) There exists an orthonormal basis

for

such that

is an orthonormal basis for

.

(e)

for all

.

Every time I start into this, it turns into something horribly messy. I feel like there's a trick I'm missing. Any thoughts? Or perhaps a full solution?

Thanks!