When eigenvectors are real, they are normalized to 1. How are complex eigenvectors normalized?
If there is at least one floating point number (real or complex) somewhere in the matrix
A: Yes, vectors are normalized (with respect to the standard Hermitian inner product
#1.Conjugate[#2] &). They are also orthogonalized if eigenspaces of
A are known to be orthogonal, e.g. when a represents a normal operator.
With only exact or symbolic entries: No.
The reason for this is that (ortho)normalization is rather inexpensive when performed in floating point numbers, but with entirely exact or symbolic expressions, it has the tendency to bloat the expressions so much that they cannot be handled anymore.