A certain kind of rank $2$ matrices can be written as a sum of two rank $1$ matrices of the same kind. I want to prove that a rank $2$ matrix of the form
$$
\left( \begin{array}{ccc}
X_{0} & X_{1} & X_{2} \\
X_{1} & X_{2} & X_{3} \\
X_{2} & X_{3} & X_{4}\end{array} \right)\in\mathcal{M}_{3\times 3}(\mathbb{C})
$$
can be written as a linear combination of two rank $1$ matrices of the same form. It is well known that a rank $k$ matrix can be written as a sum of $k$ rank $1$ matrices, but I want to prove that, in this case, they have the same form that the original matrix.
I do not know if it is true, but I think so.
 A: Let us show that $A=
\left( \begin{array}{ccc}
0 & X_{1} & 0 \\
X_{1} & 0 & 0 \\
0 & 0 & 0\end{array} \right)
$, $X_1\neq 0$, can not be writen as sum of two rank 1 matrices of the form $
\left( \begin{array}{ccc}
X_{0} & X_{1} & X_{2} \\
X_{1} & X_{2} & X_{3} \\
X_{2} & X_{3} & X_{4}\end{array} \right)
$.
First, any complex symmetric matrix with rank 1 can be written as $vv^t$, $v\in\mathbb{C}^3$. Thus any rank 1 matrix of this form can be written as $vv^t$. 
If $A$ is a sum of two rank 1 matrices of this form then $A=v_1v_1^t+v_2v_2^t$, such that $v_1v_1^t$ and $v_2v_2^t$ have the desired form. Notice that $v_1,v_2$ are linear independent otherwise $A$ would have rank smaller than 2.
Let $e_3^t=(0,0,1)$.
Thus $0=Ae_3=v_1v_1^te_3+v_2v_2^te_3$. Since $v_1,v_2$ are l.i. then $v_1^te_3=v_2^te_3=0$.
So $v_1v_1^t=\left( \begin{array}{ccc}
Z_{0} & Z_{1} & 0 \\
Z_{1} & Z_{2} & 0 \\
0 & 0 & 0\end{array} \right)$ and  $v_2v_2^t=\left( \begin{array}{ccc}
Y_{0} & Y_{1} & 0 \\
Y_{1} & Y_{2} & 0 \\
0 & 0 & 0\end{array} \right)$. 
Next, since $v_1v_1^t$ and $v_2v_2^t$ have that form then $Z_2=Y_2=0$. But this implies that $Z_1=Y_1=0$, otherwise $v_1v_1^t=\left( \begin{array}{ccc}
Z_{0} & Z_{1} & 0 \\
Z_{1} & 0 & 0 \\
0 & 0 & 0\end{array} \right)$ or  $v_2v_2^t=\left( \begin{array}{ccc}
Y_{0} & Y_{1} & 0 \\
Y_{1} & 0 & 0 \\
0 & 0 & 0\end{array} \right)$ would have rank 2. 
So $v_1v_1^t+v_2v_2^t=\left( \begin{array}{ccc}
Z_{0} & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0\end{array} \right)+\left( \begin{array}{ccc}
Y_{0} & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0\end{array} \right)\neq A$. This is a contradiction. So $A$ can not be written as a sum of two rank 1 matrices of that form.
