 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-1/5/iii/solution

Using part (i) and

A^{†} = - A

in the Cayley transform with argument

- A

gives

U^{†} = [(I + A)^{- 1}]^{†} (I - A)^{†} = (I - A)^{- 1} (I + A) = (I + A) (I - A)^{- 1} = U^{- 1} .

The factors commute because

I - A

and

I + A

are polynomials in the same matrix, and an invertible commuting matrix also commutes with the other factor's matrix inverse. Both inverses exist by part (ii). Thus $U$ is unitary.

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