I'm trying to prove that the matrix with entries $\left\{\frac{2x_ix_j}{x_i + x_j}\right\}_{ij}$ is positive definite for all n, where n is the number of rows/columns.
I was able to prove it for the 2x2 case by showing the determinant is always positive. However, once I extend it to the 3x3 case I run into trouble. I found a question here whose chosen answer gave a condition for positive definiteness of the extended matrix, and after evaluating the condition and maximizing it via software, the inequality turned out to hold indeed, but I just can't show it.
Furthermore, it would be way more complicated when I go to 4x4 and higher. I think I should somehow use induction here to show it for all n, but I think I'm missing something. Any help is appreciated.
Edit: Actually the mistake is mine, turns out there are indeed no squares in the denominator, so it turns out user141614's first answer is what I really needed. Thanks a lot! Should I just accept this answer, or should it be changed back to his first answer then I accept it?