Aims: This paper aims to derive an expression for the sensitivity of a polarimetric radio interferometer that is valid for all-sky observations of arbitrarily polarized sources, with neither a restriction on FoV nor with any a priori assumption regarding the polarization state of the source. We verify the resulting formula with an all-sky observation using the Murchison Widefield Array (MWA) telescope. Methods: The sensitivity expression is developed from first principles by applying the concept of System Equivalent Flux Density (SEFD) to a polarimetric radio interferometer not by computing $A_e/T_{sys}$. The SEFD is calculated from the standard deviation of the noisy flux density estimate for a target source due to system noise. Results: The SEFD for a polarimetric radio interferometer is generally not $1/sqrt{2}$ of a single-polarized interferometer as is often assumed for narrow FoV. This assumption can lead to significant errors for a dual-polarized dipole based system, which is common in low-frequency radio astronomy: up to $sim 15%$ for a zenith angle (ZA) coverage of $45^circ$, and up to $sim45%$ for $60^circ$ coverage. The worst case errors occur in the diagonal planes of the dipole for very wide FoV. This is demonstrated through theory, simulation and observations. Furthermore, using the resulting formulation, calculation of the off-zenith sensitivity is straightforward and unambiguous. Conclusions: For wide FoV observations pertinent to low-frequency radio interferometer such as the SKA-Low, the narrow FoV and the single-polarized sensitivity expressions are not correct and should be replaced by the formula derived in this paper.
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