An elementary inequality of use in testing convergence of eigenvector calculations is proven. If
e
λ
{e_\lambda }
is a unit eigenvector corresponding to an eigenvalue
λ
\lambda
of a selfadjoint operator
A
A
on a Hilbert space
H
H
, then
\[
|
(
g
,
e
λ
)
|
2
≤
‖
g
‖
2
‖
A
g
‖
2
−
(
g
,
A
g
)
2
‖
(
A
−
λ
I
)
g
‖
2
{\left | {(g,{e_\lambda })} \right |^2} \leq \frac {{{{\left \| g \right \|}^2}{{\left \| {Ag} \right \|}^2} - {{(g,Ag)}^2}}}{{{{\left \| {(A - \lambda I)g} \right \|}^2}}}
\]
for all
g
g
in
H
H
for which
A
g
≠
λ
g
Ag \ne \lambda g
. Equality holds only when the component of
g
g
orthogonal to
e
λ
{e_\lambda }
is also an eigenvector of
A
A
.