 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2022/10/a/solution

U^{†} U = I

and

V^{†} V = I

. The adjoint matrix reverses the order of multiplication, so

(U V)^{†} (U V) = V^{†} U^{†} U V = V^{†} V = I .

Thus the product is unitary. Likewise

(U V) (U V)^{†} = I

, or this follows from invertibility in finite dimension.

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