Let $\mathbf{A}$ be any $n\times n$ symmetric positive matrix ($A_{ij}>0$). It is easy to show that the solution to the following optimization problem 
\begin{align}
\max_{\mathbf{x}}~~\mathbf{x^TAx}\,\,;s.t.~~\mathbf{x}\geq \mathbf{0},~~||\mathbf{x}||_2=1
\end{align}
is given by the so-called perron vector of $\mathbf{A}$ which will the eigenvector corresponding to the largest eigen value (known as perron root). It will also turn out that the perron vector has all its entries as positive. I need to see if perron vector is still a solution if I replace $l_2$-norm with $l_1$-norm constraint. I need to know if perron vector will be a solution to 
\begin{align}
\max_{\mathbf{x}}~~\mathbf{x^TAx}\,\,;s.t.~~\mathbf{x}\geq \mathbf{0},~~||\mathbf{x}||_1=1
\end{align}
If it is not, how badly does it miss out. Is there any research on this?. This comes from an engineering problem I am working on.