Prove that the $5 \times 5$ Hilbert matrix, $H_5$, has five positive eigenvalues.
I know that $\lambda$ is an eigenvalue of $H_5$ iff $$\det (\lambda I_n - H_5) = 0$$
I computed $\lambda I_n - H_5$. Now I have to find the determinant of this and I believe this would take a really long time and that there must be an easier way of doing this.
How can I find the eigenvalues of this matrix?