  Let $w$ be a Muckenhoupt $A_2(\mathbb{R}^n)$ weight and
$L_w:=-w^{-1}\mathop\mathrm{div}(A\nabla)$ the degenerate elliptic operator on
the Euclidean space $\mathbb{R}^n$. In this article, the authors establish the
Riesz transform characterization of the Hardy space $H_{L_w}^p(\mathbb{R}^n)$
associated with $L_w$, for $w\in A_{q}(\mathbb{R}^n)$ and $w^{-1}\in
A_{2-\frac{2}{n}}(\mathbb{R}^n)$ with $n\geq 3$, $q\in[1,2]$ and
$p\in(q(\frac{1}{r}+\frac{q-1}{2}+\frac{1}{n})^{-1},1]$ if, for some
$r\in[1,\,2)$, $\{tL_w e^{-tL_w}\}_{t\geq 0}$ satisfies the weighted $L^r-L^2$
full off-diagonal estimate.
