 Eigenvalue addition in a common eigenbasis

If linear operators $A,B$ have a common eigenbasis $(v_i)$, then $A+B$ has the corresponding eigenvalues ${\alpha }_i+{\beta }_i$, counting algebraic multiplicity. This follows directly from linearity on each basis vector. The pairing is determined by the common eigenvectors; it is not the collection of all pairwise sums of arbitrary eigenvalues.