Is the following matrix whose entry is defined as an integral positive definite?

Question

For any given $\xi=(\xi_1,\xi_2,\cdots,\xi_n)\in\mathbb{R}^n$, define a matrix $M_{n\times n}$ by $$M_{i,j}:=\int_\mathbb{R}e^{-(x-\xi_i)^2}e^{-(x-\xi_j)^2}\mathcal{X}_{[-\pi,\pi]}(x)dx$$ where $\mathcal{X}_{[-\pi,\pi]}$ is the characteristic function on $[-\pi,\pi]$. And I wanna prove that matrix is positive definite. Since we can view $M_{i,j}$ as $$M_{i,j}=\big\langle e^{-(\cdot-\xi_i)^2}\mathcal{X}_{[-\pi,\pi]},e^{-(\cdot-\xi_j)^2}\mathcal{X}_{[-\pi,\pi]}\big\rangle_{L^2(\mathbb{R})},$$ now for every $c=(c_1,c_2,\ldots,c_n)\in\mathbb{R}^n$, \begin{align*} \sum\limits_{i=1}^n\sum\limits_{j=1}^nc_ic_jM_{i,j}&=\big\langle\sum\limits_{i=1}^nc_ie^{-(\cdot-\xi_i)^2}\mathcal{X}_{[-\pi,\pi]},\sum\limits_{j=1}^nc_je^{-(\cdot-\xi_j)^2}\mathcal{X}_{[-\pi,\pi]}\big\rangle_{L^2(\mathbb{R})}\\ &=\left\|\sum\limits_{i=1}^nc_ie^{-(\cdot-\xi_i)^2}\mathcal{X}_{[-\pi,\pi]}\right\|_{L^2(\mathbb{R})}^2\ge0 \end{align*} Thus, the matrix $M$ is supposed to be positive definite. Then I calculate $M_{i,j}$ in the error function "ERF" form and use Matlab to generate that matrix for some $\xi$. But I find there are some negative eigenvalues for that matrix which means it is not positive definite. Well, I don't know whether the proof is not valid or the numerical experiment is wrong. Any discussion and suggestion is welcome! Thanks in advance!