matrix - eigenvector bug?

Bug introduced in 7.0.1 or earlier and fixed in 10.0.0

I have a fairly simple $3\times3$ complex matrix, $$ M=\left( \begin{array}{ccc} \frac{7}{2}-\frac{i}{2} & -1+i & \frac{1}{2}+\frac{5 i}{2} \\ -1+i & 5+i & -1+i \\ \frac{1}{2}+\frac{5 i}{2} & -1+i & \frac{7}{2}-\frac{i}{2} \\ \end{array} \right) $$ Solving for the eigenvalues and eigenvectors:

mat = {{7/2 - I/2, -1 + I, 1/2 + 5 I/2}, {-1 + I, 
    5 + I, -1 + I}, {1/2 + 5 I/2, -1 + I, 7/2 - I/2}}; 
{vals, vecs} = Eigensystem[mat]

gives (Mathematica 9.0.1)

\begin{array}{ccc} 6 & 3+3 i & 3-3 i \\ \{1,-2,1\} & \left\{\frac{17}{13}+\frac{6 i}{13},0,1\right\} & \{-1,0,1\} \\ \end{array}

But that 2nd eigenvector is not right; it should be $(1,1,1)$. Is there a bug, or am I being incredibly obtuse?

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