Let $A \in \mathbb{R}^{n \times n}$ be a normalized non-strict column diagonally dominant matrix, that is:

$$a_{j,j} = \sum_{i \ne j} \left|a_{i,j}\right|$$

where $0 \le a_{j,j} \le 1$ and $-1 \le a_{i,j} \le 0$ for all $i \ne j$. Is it possible to find a symmetric, positive definite matrix $S$ such that $\langle A x, x \rangle \le \langle S x, x \rangle$ for all $x \in \mathbb{R}^n?$