# Proving the matrix is positive semidefintie through integrals

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 $$F_i, F_j$$?

• 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. Jul 26, 2021 at 13:41
• The answer to the first question that the integral will give you a positive semidefinite matrix Jul 26, 2021 at 17:36