The Eye: what was it?

[Inziladun]

I found what appears to be an interesting reference to the symbolic Eye of Sauron, in an unexpected place in the books. While Frodo was in Rivendell awaiting the start of the second phase of his quest, he looked at the sky and saw something somewhat ominous.

But low in the South one star shone red. Every night, as the Moon waned again, it shone brighter and brighter. Frodo could see it from his window, deep in the heavens, burning like a watchful eye that glared above the trees on the brink of the valley.

FOTR

'Red' and 'watchful eye'? And he sees it in the South. Surely that isn't just a coincidence. Even in Rivendell, is he being influenced by the Ring and his ordeal with the Nazgûl to see the Eye?