  Equations are presented which efficiently update or downdate the covariance
matrix of a large number of $m$-dimensional observations. Updates and downdates
to the covariance matrix, as well as mixed updates/downdates, are shown to be
rank-$k$ modifications, where $k$ is the number of new observations added plus
the number of old observations removed. As a result, the update and downdate
equations decrease the required number of multiplications for a modification to
$\Theta((k+1)m^2)$ instead of $\Theta((n+k+1)m^2)$ or $\Theta((n-k+1)m^2)$,
where $n$ is the number of initial observations. Having the rank-$k$ formulas
for the updates also allows a number of other known identities to be applied,
providing a way of applying updates and downdates directly to the inverse and
decompositions of the covariance matrix. To illustrate, we provide an efficient
algorithm for applying the rank-$k$ update to the LDL decomposition of a
covariance matrix.
