Assume that we have a full-rank diagonally dominant matrix $A$, all the diagonal elements of which are positive, all the non-diagonal elements are negative, and the sum of the absolute values of the non-diagonal elements is equal to the diagonal element. More precisely: \begin{equation} A=[a_{i,j}] \qquad a_{i,i}>0 \qquad a_{i,j} \leq 0 \textrm{ for $i \neq j$} \quad \textrm{and } \quad a_{i,i}=\sum _{j\neq i}|a_i,_j | \end{equation}

We also have a positive diagonal matrix $D$ whose trace is constant and equal to $K$: \begin{equation} d_{ij}=0 \textrm{ for } i \neq j \qquad d_{ii} \geq 0 \quad \textrm{and } \quad K=\sum_{i}d_{i,i} \end{equation}

What is the matrix $D$ such that the smallest eigenvalue of the matrix sum $T=A+D$ is as large as possible? In other words, how can we find the $d_{ii}$ 's such that we maximize the smallest eigenvalue of $T=A+D$? Thank you all in advance! :-)