To create a matrix where the ratio between the first two eigenvectors $\frac{\lambda_1}{\lambda_2}$ is large, I can set a matrix $A = \lambda_1 u u^T + \lambda_2 v v^T$ with orthonormal $u,v$ and the $\lambda_1, \lambda_2$ I want.
I now also want each row of this matrix to sum up to the same value. Setting $A = \lambda_1 u u^T + \lambda_2 v v^T$ and then manually changing, say, one element of each row does not preserve the eigenvalue condition I need. Any solutions or directions on how to maintain the condition $\lambda_1 >> \lambda_2$ while also maintaining equal row sums of the matrix would be appreciated.
Getting sums in all rows of matrix $A$ is the same as multiplying it to vector $e=(1,1,\ldots1)^T$. If all sums are the same, then: $$ Ae=\begin{pmatrix}S\\\vdots\\S\end{pmatrix} = Se. $$
So $e$ is eigenvector of matrix $A$. In that case, it's enough to take $u=e$ to guarantee this property.