Commutativity between diagonal and unitary matrices? Quick questions: 


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*if you have a diagonal matrix $A$ and a unitary matrix $B$. Do $A$ and $B$ commute?

*if $A$ and $B$ are positive definite matrices. if $a$ is an eigenvalue of $A$ and $b$ is an eigenvalue of $B$, does it follow that $a+b$ is an eigenvalue of $A+B$?

Perhaps you know of so-called normal matrices, those complex matrices $M$ that commute with their transpose conjugate $M^{\dagger}$?
If the answer to your first question were yes, there would be only "trivial", i.e. diagonal, normal matrices. Indeed, any normal matrix is unitarily equivalent to a diagonal one (all normal endomorphisms of a hermitian vector space are diagonalisable in an orthonormal basis).
