  The dynamic matrix method addresses the Landau-Lifshitz-Gilbert (LLG)
equation in the frequency domain by transforming it into an eigenproblem.
Subsequent numerical solutions are derived from the eigenvalues and
eigenvectors of the dynamic matrix. In this work we explore discretization
methods needed to obtain a matrix representation of the dynamic operator, a
fundamental counterpart of the dynamic matrix. Our approach opens a new set of
linear algebra tools for the dynamic matrix method and expose the
approximations and limitations intrinsic to it. Moreover, our discretization
algorithms can be applied to various discretization schemes, extending beyond
micromagnetism problems. We present some application examples, including a
technique to obtain the dynamic matrix directly from the magnetic free energy
function of an ensemble of macrospins, and an algorithmic method to calculate
numerical micromagnetic kernels, including plane wave kernels. We also show how
to exploit symmetries and reduce the numerical size of micromagnetic
dynamic-matrix problems by a change of basis. This procedure significantly
reduces the size of the dynamic matrix by several orders of magnitude while
maintaining high numerical precision. Additionally, we calculate analytical
approximations for the dispersion relations in magnonic crystals. This work
contributes to the understanding of the current magnetization dynamics methods,
and could help the development and formulations of novel analytical and
numerical methods for solving the LLG equation within the frequency domain.
