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?



Norm numerical range
If $A=R^\*R$ is the Cholesky factorization of $A$, then $$\frac{(ABv,v)}{(Av,v)} = \frac{(R^\*RBv,v)}{(R^\*Rv,v)} = \frac{(RBv,Rv)}{(Rv,Rv)} = \frac{(RBR^{-1}w,w)}{(w,w)}$$ for $w=Rv$, hence $W\_A(B) = W(RBR^{-1})$ and has "all the properties" of a numerical range.


