(ABRIDGED) We present here the results from new Very Long Baseline Array observations at 1.6 and 5 GHz of 19 galaxies of a complete sample of 21 UGC FRI radio galaxies. New Chandra data of two sources, viz., UGC00408 and UGC08433, are combined with the Chandra archival data of 13 sources. The 5 GHz observations of ten core-jet sources are polarization-sensitive, while the 1.6 GHz observations constitute second epoch total intensity observations of nine core-only sources. Polarized emission is detected in the jets of seven sources at 5 GHz, but the cores are essentially unpolarized, except in M87. Polarization is detected at the jet edges in several sources, and the inferred magnetic field is primarily aligned with the jet direction. This could be indicative of magnetic field shearing due to jet-medium interaction, or the presence of helical magnetic fields. The jet peak intensity $I_ u$ falls with distance $d$ from the core, following the relation, $I_ upropto d^a$, where $a$ is typically -1.5. Assuming that adiabatic expansion losses are primarily responsible for the jet intensity dimming, two limiting cases are considered: [1] the jet has a constant speed on parsec-scales and is expanding gradually such that the jet radius $rpropto d^0.4$; this expansion is however unobservable in the laterally unresolved jets at 5 GHz, and [2] the jet is cylindrical and is accelerating on parsec-scales. Accelerating parsec-scale jets are consistent with the phenomenon of magnetic driving in Poynting flux dominated jets. Chandra observations of 15 UGC FRIs detect X-ray jets in nine of them. The high frequency of occurrence of X-ray jets in this complete sample suggests that they are a signature of a ubiquitous process in FRI jets.
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