 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-1/6/a/i/solution

Using

(A B)^{†} = B^{†} A^{†}

, the conjugate transpose of the proposed matrix is

H^{†} = - i (B^{†} A^{†} - A^{†} B^{†}) = - i (B A - A B) = i (A B - B A) = H .

Thus

i [A, B] is Hermitian .

i

changes it into a Hermitian matrix. No simplicity assumption on the spectra is needed for this part.

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