TITLE: Showing that $(A_{ij})=\left(\frac1{1+x_i+x_j}\right)$ is positive semidefinite
QUESTION [7 upvotes]: Consider the matrix $A$ whose coefficients are $A_{ij} = \frac{1}{1+x_i+x_j} $ where we have 
$ x_i \geq 0$ and $ x_j \geq 0$  for $ i,j=1,2,\dots,n$. 
How can I prove that this matrix is positive semidefinite for arbitrary $n$? Using first principal minors, I proved that this is positive semi-definite for $n=2$ but I could not generalize it.

REPLY [2 votes]: Let $X= \left( \begin{array}{c} a_1 \\ \vdots \\ a_n \end{array} \right)$. Then $$\begin{array}{ll} ^tXAX & = \sum\limits_{1 \leq i,j \leq n} \frac{a_ia_j}{1+x_i+x_j} = \sum\limits_{1 \leq i,j \leq n} a_ia_j \int_0^1 t^{x_i+x_j}dt \\ & = \sum\limits_{1 \leq i,j \leq n} a_ia_j \int_0^1 t^{x_i}dt \int_0^1 t^{x_j}dt = \left( \sum\limits_{i=1}^n a_i \int_0^1t^{x_i}dt \right)^2 \geq 0 \end{array}$$