Past exam of the mathematical methods I course of the University of Oxford / 2014 resit / 5 / Solution

The characteristic polynomial of the Hermitian matrix isSolving the two eigenvector equations and using the Hermitian inner product to normalize givesIndeed $v_1^{†}{v}_2=0$ and both norms equal one. Multiplication of either eigenvector by any unit phase factor gives an equally valid normalized choice.