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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?

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    $\begingroup$ "all the same properties" what are you referring to ? $\endgroup$
    – Surb
    Commented Sep 1, 2017 at 13:52
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    $\begingroup$ @Surb First, thanks for your answer! I am thinking of properties like convexity, closedness, infinite curvature in the boundary if eigenvalues are in it, numerical range of a sum included in the sum of numerical ranges. Let $0 \notin W_A(B)$, the spectrum of $B^{-1}C$ or of $CB^{-1}$ is in $\frac{W_A(B)}{W_A(C)}$. Let $B$ be hermitian positive definite, then the spectrum of $BC$ is in $W_A(C) W_A(B)$... $\endgroup$
    – Astor
    Commented Sep 1, 2017 at 13:57

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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.

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  • $\begingroup$ Simple and frankly beautiful answer. Sorry if the question was a bit stupid. $\endgroup$
    – Astor
    Commented Sep 1, 2017 at 14:06

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