Fundamental Representation of $SU(3)$ is a complex representation

Let in a $D(R)$ dimensional representation of $SU(N)$ the generators, $T^a$s follow the following commutation rule:
$\qquad \qquad \qquad [T^a_R, T^b_R]=if^{abc}T^c_R$.

Now if $-(T^a_R)^* = T^a_R$, the representation $R$ is real. Again if we can find a unitary matrix, $V(\neq I)$ such that

$\qquad \qquad \qquad -(T^a_R)^*=V^{-1} T^a_R V \quad \forall a$

then the representation $R$ is pseudoreal.

If a representation is neither real nor pseudoreal, the representation $R$ is complex.

Claim: One way to show that a representation is complex is to show that at least one generator matrix $T^a_R$ has eigenvalues that do not come in plus-minus pairs.

Now let us consider $SU(3)$ group. The generators in the fundamental representation are given by

$T^a =\lambda^a/2; \quad a=1,...8$,
where $\lambda^a$s are the Gell-Mann matrices. We see that $T^8$ has eigenvalues $(1/\sqrt{12}, 1/\sqrt{12}, -1/\sqrt{3} )$.

My doubt is:

According to the claim, is the fundamental representation of $SU(3)$ a complex representation?

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An element of the fundamental representation of $SU(N)$ is a $n*n$ complex matrix $M$ such that $y^\dagger M^\dagger Mx = y^\dagger x$, for every complex vectors $x, y$. –  Trimok Jun 28 at 16:39