 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2014-resit/5/solution

det (A - λ I) = (\frac{3}{2} - λ)^{2} - \frac{25}{4} = (λ - 4) (λ + 1) .

Solving the two eigenvector equations and using the Hermitian inner product to normalize gives

λ_{1} = 4, v_{1} = \frac{1}{2} (1 i); λ_{2} = - 1, v_{2} = \frac{1}{2} (1 - i) .

Indeed

v_{1}^{†} v_{2} = 0

and both norms equal one. Multiplication of either eigenvector by any unit phase factor gives an equally valid normalized choice.

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