Postive-semidefiniteness of matrix with entries $1/(a_i+a_j)$
Solution 1:
Use the hint as follows:
For each $x = (x_1, \ldots, x_n)^T \in \mathbb{R}^n$, \begin{align} & x^TMx \\ = & \sum_i\sum_j x_ix_j\frac{1}{a_i + a_j} \\ = & \sum_i\sum_j \int_0^\infty x_i x_j e^{-(a_i + a_j)t}dt \\ = & \int_0^\infty \left[\sum_i\sum_j x_i x_j e^{-(a_i + a_j)t} \right] dt \\ = & \int_0^\infty \left[\sum_i\sum_j x_ie^{-a_i t}x_je^{-a_jt} \right] dt \\ = & \int_0^\infty \left(\sum_k x_ke^{-a_kt}\right)^2 dt \\ \geq & 0. \end{align} Thus $M$ is positive-semidefinite.