2 of 2 added 26 characters in body

[This is what Wolfram tech support told me when I filed a bug report:]

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}}]