 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-106/5/1/solution

h = \frac{x + x ^{*}}{2}, k = \frac{x - x ^{*}}{2 i} .

Both are hermitian and

x = h + ik

. If also

x = h^{'} + i k^{'}

with hermitian

h^{'}, k^{'}

, taking adjoints and then adding or subtracting the two equations gives

h = h^{'}

and

k = k^{'}

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