Ade, P. A. R.Ahmed, Z.Aikin, R. W.Alexander, KateBarkats, DenisBenton, S. J.Bischoff, ColinBock, J. J.Bowens-Rubin, RachelBrevik, J. A.Buder, IBullock, E.Buza, VictorConnors, Jake AnthonyCrill, B. P.Duband, L.Dvorkin, CoraFilippini, J. P.Fliescher, S.Grayson, J.Halpern, M.Harrison, SarahHildebrandt, S. R.Hilton, G. C.Hui, H.Irwin, K. D.Kang, J.Karkare, Kirit SukritKarpel, E.Kaufman, J. P.Keating, B. G.Kefeli, S.Kernasovskiy, S. A.Kovac, JohnKuo, C. L.Leitch, E. M.Lueker, M.Megerian, K. G.Namikawa, T.Netterfield, C. B.Nguyen, H. T.O’Brient, R.IV, R. W. OgburnOrlando, A.Pryke, C.Richter, S.Schwarz, R.Sheehy, C. D.Staniszewski, Z. K.Steinbach, B.Sudiwala, R. V.Teply, G. P.Thompson, K. L.Tolan, J. E.Tucker, C.Turner, A. D.Vieregg, A. G.Weber, A. C.Wiebe, D. V.Willmert, J.Wong, CWu, W. L. K.Yoon, K. W.2016-08-232016Ade, P. A. R., Z. Ahmed, R. W. Aikin, K. D. Alexander, D. Barkats, S. J. Benton, C. A. Bischoff, et al. 2016. “Bicep2/keck Array. Vii. Matrix Basede/bseparation Applied to Bicep2 and the Keck Array.” The Astrophysical Journal 825 (1) (June 30): 66. doi:10.3847/0004-637x/825/1/66. http://dx.doi.org/10.3847/0004-637X/825/1/66.1538-4357http://nrs.harvard.edu/urn-3:HUL.InstRepos:27867249A 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 ΛCDM E-modes to a level corresponding to a tensor-to-scalar ratio ofen-UScosmic background radiationcosmology: observationsgravitational wavesinflationpolarizationpixeltensorharmonicspherecovarianceBICEPB-modepolarization: linearJournal Article2016-08-2310.3847/0004-637X/825/1/66