let $x^\prime$ be a $1 \times n$ vector , $\Pi$ be a $n\times n$ matrix,
and $\mathbf{1}^\prime\ $ be a $1 \times n$ vector of 1's. When is it true
that $x^\prime \Pi \mathbf{1}+ \mathbf{1}^\prime \Pi^\prime x = 2x^\prime \Pi \mathbf{1}$. For example,if $\Pi$ is a $2 \times 2$ symmetric matrix then:
$
[x_{1} , x_{2}]
\begin{bmatrix}
\pi_{1} & \pi_{2} \\
\pi_{2} & \pi_{3}
\end{bmatrix}
\begin{bmatrix}
1 \\
1
\end{bmatrix}
+
[1 , 1]
\begin{bmatrix}
\pi_{1} & \pi_{2} \\
\pi_{2} & \pi_{3}
\end{bmatrix}
[x_{1} , x_{2}]^\prime
= 2 x_1 (\pi_1 + \pi_2) + x_2 (\pi_2 + \pi_3) = 2x^\prime \Pi \mathbf{1}
$
But in general ? Thanks