An inequality of use in testing convergence of eigenvector calculations is improved. If
e
λ
{e_\lambda }
is a unit eigenvector corresponding to an eigenvalue
λ
\lambda
of a dominant operator
A
A
on a Hilbert space
H
H
, then
\[
|
(
g
,
e
λ
)
|
2
≤
|
|
g
|
|
2
|
|
A
g
|
|
2
−
|
(
g
,
A
g
)
|
2
|
|
(
A
−
λ
I
)
g
|
|
2
|(g,{e_\lambda }){|^2} \leq \frac {{||g|{|^2}||Ag|{|^2} - |(g,Ag){|^2}}}{{||(A - \lambda I)g|{|^2}}}
\]
for all
g
g
in
H
H
for which
A
g
≠
λ
g
Ag \ne \lambda g
. The equality holds if and only if the component of
g
g
orthogonal to
e
λ
{e_\lambda }
is also an eigenvector of
A
A
. This result is an improvement of Bernstein’s result for selfadjoint operators.