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For Hermitian matrices $A,B \in \mathbb{C}^{n \times n}$, can one readily compute a set of cones that separate the maxima of $$\frac{x'Ax}{x'Bx}$$ among $x$ with unit-norm components?


i.e. where do cubics $x\mapsto(x'Bx)Ax$ and $x\mapsto (x'Ax)Bx$ intersect?

I get the sense that a more general question than mine is solved.

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    $\begingroup$ Do you agree with my edits? $\endgroup$ – Rodrigo de Azevedo Apr 19 at 20:28
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In case of intersection, necessarily, there is $\lambda$ s.t. $Ax=\lambda Bx$; then $\lambda$ is a generalized eigenvalue of the matrix-bundle $(A,B)$ and $x$ is an eigenvector associated to $\lambda$. Since $A,B$ are hermitian, this $\lambda$ is real and $x$ may be a non-real vector.

In these conditions $x^*BxAx=x^*AxBx$ for all such eigenvectors.

In Maple, the eigen-elements are given by the command $evalf(Eigenvectors(A,B))$.

$\textbf{Remark.}$ Usually, we also consider the bundle $(B,A)$ but here, it suffices to add the vectors of the kernel of $B$.

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