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I have what I think is a simultaneous eigenvalue problem in three parameters:

$$\alpha A_1x + \beta B_1x + \gamma C_1x + D_1x = 0$$ $$\alpha A_2x + \beta B_2x + \gamma C_2x + D_2x = 0$$ $$\alpha A_3x + \beta B_3x + \gamma C_3x + D_3x = 0$$

The matrices $A_1$,$B_1$ etc are square and may be singular. I want to solve for the vector x and the scalar parameters $\alpha$, $\beta$ and $\gamma$.

I think this is a simultaneous eigenvalue problem, but other articles seems to talk about a different $x$ for each equation, where I have a shared $x$.

Does anyone know how to solve this type of problem? Are there libraries available to solve simultaneous eigenvalue problems?

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Why is this an eigenvalue problem? –  copper.hat Oct 25 '12 at 6:56

1 Answer 1

This problem is a subset of the problem called a multiparameter eigenvalue problem. The solution can be explicitly expressed as the solution to a standard (or generalized) eigenvalue problem using Kronecker products. Considerable theory was worked out by Atkinson in the 60's, but more recently popularized in numerical methods by for instance Plestenjak and Hochstenbach. The Kronecker product reformulation is for instance described on page 2 in

You have a special case, since all three equations share the same eigenvector. It should however come out as one solution using any numerical or theoretical technique for multiparameter eigenvalue problems.

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