Past exam of the mathematics course of the University of Cambridge / 2008 / ia / Paper 1 / 2A / i

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- Solution i

Solution

By the unitary matrix identities,

U^{†} U U U^{†} = (A - i B) (A + i B) = A^{2} + B^{2} + i (A B - B A) = I, = (A + i B) (A - i B) = A^{2} + B^{2} - i (A B - B A) = I .

Subtracting the two equations yields

2 i (A B - B A) = 0

. Therefore

A B = B A .

The subtraction is important: entrywise real and imaginary parts cannot generally be used to separate these terms, since the Hermitian matrices need not have real entries.

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