# Multiparameter eigenvalue problem

I am mostly asking about terminology here.

Background A generalized linear eigenvalue problem is of the form $Ax=\lambda B x$, where $A$ and $B$ are square matrices, $\lambda$ is the eigenvalue and $x$ is the eigenvector. We can write this in the form that makes it evident that it is linear in $\lambda$: $(A-\lambda B) x = 0$. A polynomial eigenvalue problem is of the form $$\left[\sum_{k=0}^N \lambda^k A_k\right] x = 0$$

Question Now, I am wondering what you would call an eigenvalue problem which is "bilinear" in its eigenparameters: $$\left[ A\lambda\mu + B\lambda + C\mu + D \right] x = 0$$ where $\lambda$ and $\mu$ are the eigenvalues, $x$ is the eigenvector and $A,B,C,D$ are square matrices.

Has this kind of problem been studied? I know this is generally a nonlinear eigenvalue problem of the form $M(\lambda,\mu)x = 0$, but it seems like there is important additional structure here.

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Here, a more general quadratic multiparameter eigenproblem is considered. Here is a related paper. – J. M. Dec 18 '10 at 10:54
@J. M.: Wow, I didn't realize you can linearize the problem like that. Thanks for the reference. – Victor Liu Dec 18 '10 at 21:28