Eigenpair ($(\lambda ,v),Av=\lambda v,v\ne 0$)

An eigenpair consists of an eigenvalue and a nonzero corresponding eigenvector of a specified linear operator. Scaling the eigenvector by a nonzero scalar gives another representation of the same eigendirection. For an approximate pair $(\hat{\lambda },\hat{v})$, its eigenpair residual is $A\hat{v}-\hat{\lambda }\hat{v}$; a small residual alone need not give a small eigenvector error without spectral separation and conditioning information.