negative variance example I came across the following matrix.

I notice, that it is symmetric and real and has an eigenvalue of -0.01091786.
This must mean, that there exists a vector $x \in \mathbb{R}^{10}$ for which it holds
$$ xVx^T<0$$
but I have no idea how to come about an example of such vector? - I especially want to know, if there is an example of a vector $x$ that has that property and also has the property $\sum_{i=1}^{10}x_i =1$.
Thank you in advance.

Here is one trick for finding eigenvectors associated with an eigenvalue. It is guaranteed to work if the characteristic polynomial is also the minimal polynomial, but may fail otherwise:
If $p$ is the characteristic polynomial of $A$, then by the Cayley-Hamilton theorem, $p(A) = 0$. If $\lambda$ is an eigenvalue of $A$, then $p(t) = (t - \lambda)q(t)$ where the polynomial $q(t)$ can be found by synthetic division.
Now $p(A) = (A - \lambda)q(A) = 0$. If $q(A)$ has any non-zero columns, those columns have to be eigenvectors associated with $\lambda$.
