Within the framework of three lepton families we have derived an exact and compact formula to describe the matter effect on neutrino mixing and CP violation. This model- and parametrization-independent result can be particularly useful for recasting the fu

predictionsforVwiththeexperimentaldataofneutrinooscillations.Thelattermayinmostcasesbeinvolvedinthepotentialmattere ectsandmustbecarefullyhandled.Inmatterthee ectiveHamiltonianforneutrinoscanbewrittenas[6]Hν=1

2GFNeEdescribesthecharged-

currentcontributiontotheνee forwardscattering,Neisthebackgrounddensityofelectrons,andEstandsfortheneutrinobeamenergy.Theneutral-currentcontributions,whichareuniversalforνe,νµandντneutrinos,leadonlytoanoverallunobservablephaseandhavebeenneglected.TransformingHνfromthe avoreigenbasistothemasseigenbasis,weobtainthee ectivemass-squaredmatrixofneutrinos:

2m2

1+A|Ve1| ν= AVe1Ve 2 AVe1Ve3 AVe 1Ve22m22+A|Ve2|AVe2Ve 3

ThisHermitianmatrixcanbediagonalizedthroughaunitarytransformation:U νU=Diag{λ1,λ2,λ3},whereλidenotethee ectivemass-squaredeigenvaluesofneutrinos.Explicitlyone nds

λ1=m2

1+1

1

133

3 AVe 1Ve3 AVe V ,e322m23+A|Ve3| (4)3(1 z2),3(1 z2), λ2=m21+λ3=m2

1+z

3arccos222x3 9xy+27A m2

21 m31|Ve1|

Di,Uij=A

TheexpressionsofλiinRef.[7]haveapparentprintingerrors,whilethosegiveninRef.[10]dependuponaspeci cparametrizationofV.