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This came up as the solution to a constrained optimization problem (after doing some analysis to find the stationary points of some Lagrangian): $$ A_{ij} \exp(\eta_j) = B_{ij} \exp(\eta_i) $$ $A$,$B$ are square of dimension $n\times n$. $B$ is known and we want to find $A$ and $\eta$ (the $n$-dim vector of lagrange multipliers).

Is there an analytic solution?

MrYouMath
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tangerine
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  • Looks like you can define $\eta_i$ for all $i$ any way you like, then let $A_{ij} = B_{ij} e^{\eta_i - \eta_j}$ for all $i,j$. – Michael Dec 18 '17 at 09:42
  • I assume that your equation is element-wise and there is not an implied summation over repeated indices. You have $n^2+n$ unknowns and $n^2$ equations so you need some more information which is presumably the constraints that generated the LMs – user121049 Dec 18 '17 at 10:47

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