# Solving Multiparameter Eigenvalue problem

I have a multiparameter polynomial eigenvalue problem of the form:

$$(\alpha^2 A + \alpha\beta B + \alpha\gamma C + \alpha D + \beta^2 E + \beta\gamma F + \beta G + \gamma^2 H + \gamma J + K)\cdot{t} = 0$$

where $\alpha$, $\beta$, and $\gamma$ are unknown scalars, and A...K are 4x4 matrices.

We can linearize it to:

$$\left( \alpha \begin{bmatrix} A & B & C & D \\ 0 & 0 & 0 & I \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} + \beta \begin{bmatrix} 0 & E & F & G \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & I \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} + \gamma \begin{bmatrix} 0 & 0 & H & J \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & I \\ \end{bmatrix} + \begin{bmatrix} 0 & 0 & 0 & K \\ -I & 0 & 0 & 0 \\ 0 & -I & 0 & 0 \\ 0 & 0 & -I & 0 \\ \end{bmatrix} \right) \begin{bmatrix} \alpha t \\ \beta t \\ \gamma t \\ t \end{bmatrix} = 0,$$ which we will rewrite as, $$(\alpha X + \beta Y + \gamma Z + W) \hat{t} = 0$$

where $\hat{t} = [\alpha t^\top \; \beta t^\top \; \gamma t^\top \; t^\top]^\top$. This is a multiparameter eigenvalue problem. However, I have not been able to find literature for methods to readily solve this (there is plenty of literature for 2-parameter, but I cannot find > 2 parameters).

Could anybody point me to relevant methods or papers that could explain how to solve this?

Have you tried computing $\det(\alpha^2 A + \ldots + K)$? It will be a (possibly rather unpleasant) polynomial in $\alpha, \beta, \gamma$, and there is a nontrivial $t$ when it is $0$.
Could you be more precise about what you want to compute? The solutions $(\alpha,\beta,\gamma)$ and $t$ to the first equation do not form a countable set, since you can for instance fix (arbitrary) $\beta$ and $\gamma$ and get $2n$ solution pairs $(\alpha,t)$.