# Localizing the intersections of cubics

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.

• Do you agree with my edits? – Rodrigo de Azevedo Apr 19 at 20:28

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 $$B$$.