Eigenvalue addition in a common eigenbasis ($A{v}_i={\alpha }_i{v}_i,B{v}_i={\beta }_i{v}_i⟹(A+B)v_i=({\alpha }_i+{\beta }_i)v_i$)

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.