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I have come across quite a lot of proofs to show that a matrix is positive semidefinite through integrals. Examples are

Showing that $(A_{ij})=\left(\frac1{1+x_i+x_j}\right)$ is positive semidefinite

Proving positive definiteness of matrix $a_{ij}=\frac{2x_ix_j}{x_i + x_j}$

Prove positive definiteness

It seems like, a common trait is writing the matrix entries as an integral of a dot product $M_{ij} = \int_a^b F_i(x) F_j(x) g(x) dx$. Then one can kind of think of this matrix as $M = B^T B$. Here $B$ has "infinite rows" as instead of summation we are doing integration, but a lot of ideas can be borrowed.

Is my understanding correct? Is there a name for this kind of approach and matrix? Is there a trick to find the $F_i, F_j$?

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    $\begingroup$ I'm not sure about the infinite matrix. I would think of this more akin to the fact that $\sum_i B_i^T B_i$ is positive semidefinite as each $B_i^T B_i$ is positive semidefinite, except you replace the sum with an integral. $\endgroup$
    – Rammus
    Jul 26, 2021 at 13:41
  • $\begingroup$ The answer to the first question that the integral will give you a positive semidefinite matrix $\endgroup$ Jul 26, 2021 at 17:36

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