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Take two matrices $A,B \in \mathbb{R}^{n\times n}$. I am considering the generalized eigenvalue problem ($\gamma \in \mathbb{R}, \phi \in \mathbb{R}^n$)

$$A\phi = \gamma B\phi.$$

Is there a systematic way of modifying the matrix $B$ such that $\phi$ acquires desired properties?

I have a specific example in mind: let $n=4m$ and require $\phi$ to be in the m-fold product of the unit sphere in $\mathbb{R}^4$, i.e. when $\phi$ is chopped into blocks of 4 entries, these vectors each have Euclidean norm 1.

For a fixed $A\in GL(n)$, can one construct a diagonal matrix $B$ such that there exists an eigenpair $(\gamma,\phi)$ with the desired property on $\phi$?

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  • $\begingroup$ Your exact goal is unclear to me. I presume you want to find the pairs $(\phi,\gamma)$ that solve your equation, but do you want to do it for all $A$ and $B$? Or for some nice fixed $B$ and all $A$? What modifications of $B$ do you allow? I assume "replace $B$ with $I$" is not the kind of systematic modification you have in mind. $\endgroup$
    – Joonas Ilmavirta
    Aug 26, 2014 at 10:51
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    $\begingroup$ @JoonasIlmavirta Wiki: generalized eigenproblem should make the setting clearer and answer your first two questions. I second the third one though: what kind of modifications? What kind of properties? This looks too general; can you be more specific or make an example? $\endgroup$
    – Federico Poloni
    Aug 26, 2014 at 11:03
  • $\begingroup$ I added a specific example with a rather complicated condition on $\phi$. $\endgroup$
    – madison54
    Aug 26, 2014 at 11:34

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(Probably this is not what you are asking for, but it might be a cue to modify your question in the right direction.)

If you fix $A$, $\gamma$, $\phi$, then what you get is a linear system in the entries of $B$.

If $B$ has to be diagonal, then you have $n$ equations and $n$ unknowns, and it is easy to see that it is solvable whenever $\gamma\neq 0$ and $\phi$ has no zero entries.

If $B$ has a more general structure, then you may end up with an underdetermined system, or with a nonlinear one, but there are algorithm for those as well.

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