  A linear polarization field on the sphere can be uniquely decomposed into an
E-mode and a B-mode component. These two components are analytically defined in
terms of spin-2 spherical harmonics. Maps that contain filtered modes on a
partial sky can also be decomposed into E-mode and B-mode components. However,
the lack of full sky information prevents orthogonally separating these
components using spherical harmonics. In this paper, we present a technique for
decomposing an incomplete map into E and B-mode components using E and B
eigenmodes of the pixel covariance in the observed map. This method is found to
orthogonally define E and B in the presence of both partial sky coverage and
spatial filtering. This method has been applied to the BICEP2 and the Keck
Array maps and results in reducing E to B leakage from LCDM E-modes to a level
corresponding to a tensor-to-scalar ratio of $r<1\times10^{-4}$.
