A Hilbert matrix $A$ is a square $n\times n$ matrix with entries given by $$a_{ij} = \int^1_0 t^{i-1}t^{j-1}dt = \frac{1}{i+j-1}$$.
Prove that for each $n$. the Hilbert matrix $A$ is positive definite.
Here are my workings,
Let $X=(x_i)_{1\leq i\leq n} \in \cal{M}_{n,1}(\mathbb{R}).$ We have $$ {}^tXAX=\sum_{1\leq i,j\leq n}\frac{x_ix_j}{i+j-1}=\sum_{1\leq i,j\leq n}x_ix_j\int_0^1t^{i+j-2}dt=\int_0^1\left(\sum_{i=1}^nx_it^{i-1}\right)^2dt>0 $$ for $X\neq0$, giving the announced result since $A$ is symmetric.
Is this correct? I have been given a hint, consider the expression $x^TAx$ and usethe above equation I listed above, $a_{ij} = \int^1_0 t^{i-1}t^{j-1}dt = \frac{1}{i+j-1}$, but I am not quite sure how to use this hint, looking for some help, thanks!