 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/a/solution

A x = λ x

on the left by

x^{†}

. The unit vector normalization is

x^{†} x = 1

, so

λ = x^{†} A x = x^{†} H x + i x^{†} S x .

No normal matrix assumption on

A

is needed: this is an identity for each normalized eigenvector.

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