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Can this be set up as an eigenvalue problem?

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I think the connection here is just that both problems deal with quadratics, so the processes resemble each-other. I don't think there's any deeper direct connection to be found, unfortunately.
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The transformation isn’t linear so there’s no matrix A such that T([x,y]^(T)) = A*([x,y]^(T)) for all [x,y].

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