Let $V$ be a nonzero finite-dimensional inner product space over $\mathbb F$ (where $\mathbb F$ denotes either $\mathbb R$ or $\mathbb C$), and $\dim_{\mathbb F}(V) = n$.
The exercises asks to prove if with respect to an orthonormal basis of $V$, say $e_{1}, \ldots , e_{n}$, an operator $T$ on $V$ has a Hilbert matrix, then $T$ is positive and invertible.
Clearly $T$ is self-adjoint because the conjugate transpose of Hilbert matrix is the Hilbert matrix. I want to show that $T$ is positive and invertible by showing for all $v \in V$ and $v$ not being $0, \langle Tv, v \rangle > 0$.
I take $v = a_{1}e_{1} + \cdots + a_{n}e_{n}$ with $a_{1}, \ldots , a_{n} \in \mathbb F$ not all being $0$ and brute force the calculation of $\langle Tv, v \rangle$ and arrive at $$ \sum_{j=1}^{n}\sum_{i=1}^{n}1/(i+j-1) a_{i}\overline{a_{j}}$$ But then I am stuck at showing this is positive. What steps can I take? Or is there other simpler approach to the original problem?
