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A is a square matrix with positive elements and x is a real vector (both of them n>1 dimensional). Prove that for any such matrix and vector $$\sum\limits_{i,j = 1}^n {{A_{i,j}}({x_i}^2 - {x_i}{x_j})} \ge 0 $$ P.S.#1: I am not 100% sure that the inequality holds but I have seen it happen in numerous numerical examples (for random matrices and vectors), P.S.#2: I need this to prove the uniqueness of the solution to a PDE.

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    $\begingroup$ Is $A$ symmetric? If so, I think a diagonal dominance argument can be used to show the above inequality is true. $\endgroup$ – copper.hat Jul 19 '14 at 15:22
  • $\begingroup$ @copper.hat Unfortunately the only feature of A is that all of its elements are non-negative $\endgroup$ – Seyed Mohsen Ayyoubzadeh Jul 19 '14 at 15:31

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