Vtyjkyuk

I am trying to show that the following matrix is positive semi-definite (PSD):

$$H=left[beginarraycc
int v(x),dx & int xv(x),dx\
int xv(x),dx & int x^2v(x),dx
endarrayright]$$

where $v(x)$ is a differentiable function of unknown sign.

The definition of PSD requires that for all $w,z in mathbbR$, $$w^2 int v(x)dx + 2wzleft(int xv(x),dxright)^2+z^2int x^2v(x),dx geq 0$$

Even if I assume $v(x) geq 0$ (I have $xgeq0$), it doesn't look like Holder's inequality is useful to get to the desired result.

Alternatively, assuming that $int v(x) dx geq 0$, I tried proving that the determinant of $H$ is positive, but that did not lead anywhere either.

Are there any other avenues?

A way to verify that a matrix is PSD is the to test that the determinants of all upper-left sub-matrices are non negative.

This is true for the first sub-matrix if you assume $int v(x) dxge0$.

For the second one you need to have $$left(int v(x) dxright)left(int x^2v(x) dxright)ge left(int xv(x) dxright)^2$$ which is Cauchy-Schwarz inequality for the inner product $langle f,g rangle = int fg$ with $f(x) = sqrtv(x)$ and $g(x)=xsqrtv(x)$ providing those functions are well defined. Which is the case if $v$ is supposed non negative.