Eigenpair residual ($r=Av-\lambda v$)

For an approximate eigenpair $(\tilde{\lambda },\tilde{v})$, the residual is

For a real symmetric matrix $M$, real $\mu$ and nonzero vector $u$, some eigenvalue of $M$ lies within the stated ratio of $\mu$. In an orthonormal eigenbasis, write $u={\sum }_j{c}_j{q}_j$ and $M{q}_j={\theta }_j{q}_j$. Then $\Vert Mu-\mu u{\Vert }_2^2={\sum }_j({\theta }_j-\mu {)}^2{c}_j^2\ge {\min }_j|{\theta }_j-\mu {|}^2\Vert u{\Vert }_2^2$, proving the bound.

If $\Vert \tilde{v}{\Vert }_2=1$, the smallest operator-norm perturbation that makes $(\tilde{\lambda },\tilde{v})$ an exact eigenpair has norm $\Vert r{\Vert }_2$. One such rank-one perturbation is