= Eigenvalue addition in a common eigenbasis
{title2=$Av_i=\alpha_iv_i,\ Bv_i=\beta_iv_i\ \Longrightarrow\ (A+B)v_i=(\alpha_i+\beta_i)v_i$}
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>.
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