Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2002 / Paper 1 / 5 / i

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- Solution i

Solution

The conjugate transpose reverses products and conjugates scalars. Since

A, B

are Hermitian matrices,

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

Therefore

H is Hermitian .

In particular the commutator of two Hermitian matrices is anti-Hermitian, and multiplication by

i

converts it to a Hermitian matrix.

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