# Recasting a dispersion relation (determinant =0) as a matrix eigenvalue problem for numerical solution

I have a physical problem that involves a dispertion relation between two parameters, the frequency $$\omega$$ and the propagation coefficient $$\beta_{n}$$, where $$n$$ is an integer index. The dispersion equation, whose solution will give us the relation $$\omega=\omega(\beta_{n})$$, is deduced from the system's matrix by requiring that its determinant vanish, $$\det[ M ]=0,$$ where $$M$$ is a Hermitian matrix of order $$2(2N+1)\times 2(2N+1)$$, with $$N$$ being the highest allowed values of the index $$n$$ (basically the truncation of a sum over $$-N\leq n\leq N$$). The elements of the matrix $$M$$ include terms, such as Bessel and Hankel functions, that are functions of $$\omega$$ and $$\beta_{n}$$ (too long and convoluted to practically write here explicitly). Typically such a system is solved numerically for a given $$\omega$$ by forcing the determinant above to vanish and taking the corresponding roots as solutions of $$\beta_{n}$$.

For a given $$\omega$$, when I try to directly solve this determinant =0 equation, to find $$\beta_{n}$$, I run into numerical problems (for example in Mathematica) and so far finding the roots of this determinant directly seem to be hopeless. However, I am aware that a method like Arnoldi's method can find the eigenvalues of a given matrix quickly, and it works for my matrix $$M$$, if I test it with given $$\omega$$ and $$\beta_{n}$$ values.

However, I am not sure how to express my problem as an eigenvalue problem here, so as to take advantage of the Arnoldi method, which gives approximate eigenvalues of the matrix? How can I recast this problem so that the eigenvalues returned would give $$\beta_{n}$$? What is the relation between roots of $$\det[M]$$ and of the eigenvalues of $$M$$?