  We present a model for a Noise Matched Phased Array Feed (PAF) system and
compare model predictions with the measurement results. The PAF system consists
of an array feed, a receiver, a beamformer and a parabolic reflector. The novel
aspect of our model is the characterization of the {\em PAF system} by a single
matrix. This characteristic matrix is constructed from the open-circuit voltage
covariance at the output of the PAF due to signal from the observing source,
ground spillover noise, sky background noise and (low-noise) amplifier (LNA)
noise. The best signal-to-noise ratio on the source achievable with the PAF
system will be the maximum eigenvalue of the characteristic matrix. The voltage
covariance due to signal and spillover noise are derived by applying the
Lorentz reciprocity theorem. The receiver noise covariance and noise
temperature are obtained in terms of Lange invariants such that they are
suitable for noise matching the array feed. The model predictions are compared
with the measured performance of a 1.4 GHz, 19-element, dual-polarized PAF on
the Robert C. Byrd Green Bank Telescope. We show that the model predictions,
obtained with an additional noise contribution due to the measured losses ahead
of the low-noise amplifier, compare well with the measured ratio of system
temperature to aperture efficiency as a function of frequency and as a function
of offset from the boresight. Further, our modeling indicates that the
bandwidth over which this ratio is optimum can be improved by a factor of at
least two by noise matching the PAF with the LNA.
