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[This is what Wolfram tech support told me:]

That is because using machine precision (inexact) numbers such 0. (as opposed to exact numbers like 0) forces Eigenvectors[] to seach for eigenvectors numerically and hence the error message. For example, please evaluate and compare the results of the following expression used with machine precision and exact numbers:

Eigenvectors[{{Cos[1. x], Exp[I x]}, {1, 0}}]

Eigenvectors[{{Cos[1 x], Exp[I x]}, {1, 0}}]