Discussion
If H is a Hermitian operator, the following is "always" true :
Answer: A
It is always true that H has real diagonal matrix elements.
A Hermitian operator satisfies \(H_{ij} = H_{ji}^{*}\). Setting \(i = j\) gives \(H_{ii} = H_{ii}^{*}\), so every diagonal element is real, in any basis.
The others are not guaranteed. Diagonal elements can be negative (for example \(-\sigma_{z}\)). Off-diagonal elements are complex in general, and can be real, as in \(\sigma_{x}\). The trace is the sum of real eigenvalues and need not be zero (the identity is Hermitian).
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