 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/ia/paper-1/7a/solution

Let

v \neq = 0

A

λ

. Multiplying

A v = λ v

v^{†}

gives

λ = \frac{v ^{†} A v}{v ^{†} v} .

∥ v ∥^{2}

, while

\overline{v^{†} A v} = v^{†} A^{†} v = v^{†} A v .

Thus the numerator is real, and every eigenvalue of a Hermitian matrix is real. This argument does not require distinct eigenvalues.

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