  The Davis-Kahan theorem can be used to bound the distance of the spaces
spanned by the first $r$ eigenvectors of any two symmetric matrices. We extend
the Davis-Kahan theorem to apply to the comparison of the union of eigenspaces
of any two symmetric matrices by making use of polynomial matrix transforms and
in so doing, tighten the bound. The transform allows us to move requirements
present in the original Davis-Kahan theorem, from the eigenvalues of the
compared matrices on to the transformation parameters, with the latter being
under our control. We provide a proof of concept example, comparing the spaces
spanned by the unnormalised and normalised graph Laplacian eigenvectors for
$d$-regular graphs, in which the correct transform is automatically identified.
