#2 counterexample
Let
A=\left[\begin{array}{cc}
i & 1 \\
1 & -i
\end{array} \right]
where #i# is the imaginary value. Note that A^T=A and has the characteristic equation\lambda^2=0,
thus the eigenvalue of A is \lambda=0 with algebraic multiplicity of 2. The eigenvector of A is...