If I have some function $G(x), \quad G:X\to\mathbb{R}$, which is used to construct a matrix as follows $$ A_{ij} = G\Bigl(\frac{x_i + x_j}{2}\Bigr) $$
and I can show that the $2\times2$ matrix has all non-negative eigen values, is there a way to generalise this to an $N\times N$ matrix, perhaps via some inductive argument, to show that the matrix $A_{ij}^N$ (n indicating the size of the matrix, not an exponent) is positive semi-definite?
I understand another way to show PSD is by showing that the determinants of all the principle minors are +ve, and therefore all the pivots are non-negative, and there is a recursive argument on determinants of an $N\times N$ matrix, but this particular route seems very unwieldy. However, if this is the way to go, please could you provide details? Similarly, the are plenty of other ways of showing PSD, but I have abandoned them as unweidly (for example, the Cholesky decomposition). Thank you
