 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1/6/i/b/solution

K

x^{†} K x

is real because its complex conjugate equals

x^{†} K^{†} x = x^{†} K x

. Applying this to

H, S

in part (a) gives

Re λ = x^{†} H x, Im λ = x^{†} S x .

These are the real and imaginary quadratic forms associated with the numerical range of

A

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