  We revisit the $k$-Hessian eigenvalue problem on a smooth, bounded,
$(k-1)$-convex domain in $\mathbb R^n$. First, we obtain a spectral
characterization of the $k$-Hessian eigenvalue as the infimum of the first
eigenvalues of linear second-order elliptic operators whose coefficients belong
to the dual of the corresponding G\r{a}rding cone. Second, we introduce a
non-degenerate inverse iterative scheme to solve the eigenvalue problem for the
$k$-Hessian operator. We show that the scheme converges, with a rate, to the
$k$-Hessian eigenvalue for all $k$. When $2\leq k\leq n$, we also prove a local
$L^1$ convergence of the Hessian of solutions of the scheme. Hyperbolic
polynomials play an important role in our analysis.
