I know that the Hadamard product is distributive over addition. But suppose that $A, B \in \mathbb{R}^{n \times n}$ and $v \in \mathbb{R}^n$. Then can we say
$$v^T (A \circ B) v = v^TAv \cdot v^TBv$$
where $\circ$ denotes the Hadamard product, $\cdot$ is standard scalar multiplication, and the rest are matrix-vector multiplications?