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

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- Solution 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 .

The commutator of two Hermitian matrices is skew-Hermitian, and multiplication by

i

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

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