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 eigenvalue (known as the 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 the $2$-norm constraint with the $1$-norm constraint. I need to know if the 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.