Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2003 / Paper 1 / 5 / a

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- Solution a

Solution

Since

A

and

B

are real symmetric matrices,

U^{†} = A - i B

. Expanding the unitarity condition gives

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

Equality of the entrywise real and imaginary parts yields

A^{2} + B^{2} = I, A B = B A .

These are the identities for the real and imaginary parts of a symmetric unitary matrix.

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