Discussion
Eigenvalues of the sum of two operators is the sum of the eigenvalues if :
Answer: B
The eigenvalues add when the operators have diagonalizable commuting matrices.
If \(A\) and \(B\) are diagonalizable and commute, they can be diagonalized by the same basis. In that basis each common eigenvector \(v\) satisfies \(Av = av\) and \(Bv = bv\), so \((A + B)v = (a + b)v\): the eigenvalues of the sum are sums of corresponding eigenvalues.
Being Hermitian, unitary or anti-Hermitian is not enough. For example, the Pauli matrices \(\sigma_{x}\) and \(\sigma_{z}\) are Hermitian with eigenvalues \(\pm 1\), but \(\sigma_{x} + \sigma_{z}\) has eigenvalues \(\pm\sqrt{2}\), because they do not commute.
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