  Suppose two Hermitian matrices $A,B$ almost commute ($\Vert [A,B] \Vert \leq
\delta$). Are they close to a commuting pair of Hermitian matrices, $A',B'$,
with $\Vert A-A' \Vert,\Vert B-B'\Vert \leq \epsilon$? A theorem of H. Lin
shows that this is uniformly true, in that for every $\epsilon>0$ there exists
a $\delta>0$, independent of the size $N$ of the matrices, for which almost
commuting implies being close to a commuting pair. However, this theorem does
not specify how $\delta$ depends on $\epsilon$. We give uniform bounds relating
$\delta$ and $\epsilon$. We provide tighter bounds in the case of block
tridiagonal and tridiagonal matrices and a fully constructive method in that
case. Within the context of quantum measurement, this implies an algorithm to
construct a basis in which we can make a {\it projective} measurement that
approximately measures two approximately commuting operators simultaneously.
Finally, we comment briefly on the case of approximately measuring three or
more approximately commuting operators using POVMs (positive operator-valued
measures) instead of projective measurements.
