  In this article we unravel the role of matter effect in neutrino oscillation
in the presence of lepton-flavor-conserving, non-universal non-standard
interactions (NSI's) of the neutrino. Employing the Jacobi method, we derive
approximate analytical expressions for the effective mass-squared differences
and mixing angles in matter. It is shown that, within the effective mixing
matrix, the Standard Model (SM) W-exchange interaction only affects
$\theta_{12}$ and $\theta_{13}$, while the flavor-diagonal NSI's only affect
$\theta_{23}$. The CP-violating phase $\delta$ remains unaffected. Using our
simple and compact analytical approximation, we study the impact of the
flavor-diagonal NSI's on the neutrino oscillation probabilities for various
appearance and disappearance channels. At higher energies and longer baselines,
it is found that the impact of the NSI's can be significant in the numu to numu
channel, which can probed in future atmospheric neutrino experiments, if the
NSI's are of the order of their current upper bounds. Our analysis also enables
us to explore the possible degeneracy between the octant of $\theta_{23}$ and
the sign of the NSI parameter for a given choice of mass hierarchy in a simple
manner.
