Let $A \in \mathbb{R}^{n \times n}$ be a symmetric positive definite matrix, and let $B \in \mathbb{R}^{n \times n}$ be an arbitrary matrix.

Define the numerical range or field of values of $B$ as \begin{align} W(B) = \left\{\frac{(Bv,v)}{(v,v)}, 0 \ne v \in \mathbb{C}^n \right\} \end{align} where $(\cdot,\cdot)$ is the euclidean inner product.

Would the numerical range \begin{align} W_A(B) = \left\{\frac{(ABv,v)}{(Av,v)}, 0 \ne v \in \mathbb{C}^n \right\} \end{align} have all the same properties than $W(B)$ since the norms are equivalent?