# Showing a matrix positive semidefinite

For $k\in\Bbb{Z}$ define $y_k=\int_{-1}^1x^2e^{ikx}dx$. I have to show that the matrix $((a_{r,s}=y_{r-s}))_{1\le r,s\le n}$ is positive semidefinite for all $n\in\Bbb{N}$.

All the diagonal elements are $\frac23$. But the off diagonal elements are $a_{r,s}=\frac{e^{i(r-s)}-5e^{-i(r-s)}}{i(r-s)}$. This doesn't even make the matrix symmetric, or I am miscalculating. How do I show positive semidifinteness

• Did you include symmetry in your definition of positive definiteness? Some do, some don't... – user190080 Apr 29 '18 at 12:59

$\sum_{j=1}^{n} \sum_{j=1}^{n} c_j \overline c_k y_{(j-k)} =\int_1^{1} x^{2} |\sum_{j=1}^{n} c_j e^{ijx}|^{2}dx \geq 0$.
• Just expand $|\sum c_j e^{ijx}|^{2}$ using the formula $|z|^{2}=z\overline {z}$ – Kavi Rama Murthy Apr 30 '18 at 4:49