  Sensitivity of eigenvectors and eigenvalues of symmetric matrix estimates to
the removal of a single observation have been well documented in the
literature. However, a complicating factor can exist in that the rank of the
eigenvalues may change due to the removal of an observation, and with that so
too does the perceived importance of the corresponding eigenvector. We refer to
this problem as "switching of eigenvalues". Since there is not enough
information in the new eigenvalues post observation removal to indicate that
this has happened, how do we know that this switching has occurred? In this
paper, we show that approximations to the eigenvalues can be used to help
determine when switching may have occurred. We then discuss possible actions
researchers can take based on this knowledge, for example making better choices
when it comes to deciding how many principal components should be retained and
adjustments to approximate influence diagnostics that perform poorly when
switching has occurred. Our results are easily applied to any eigenvalue
problem involving symmetric matrix estimators. We highlight our approach with
application to a real data example.
