Let us consider a problem of the form
$$(\mathcal{L} + k^2) u(\mathbf{x})=0\, ,\quad \forall \mathbf{x} \in \Omega$$
with Dirichlet boundary conditions
$$u(\mathbf{x}) = 0, \quad \forall \mathbf{x} \in \partial\Omega\, .$$
We discretize this problem using, for example, the finite element method, to obtain the following generalized eigenvalue problem
$$[K]\{U\} = k^2 [M]\{U\} \, .$$
To impose boundary conditions one can do one of the following:
Reduce the system by deleting rows/columns corresponding to boundary degrees of freedom in matrices $[K]$ and $[M]$.
Zero row/column values for the boundary degrees of freedom of both matrices $[K]$ and $[M]$, and fix the diagonal values in $[K]$ to any finite number. The problem with this approach is that the matrix $[M]$ loses its positive-definiteness (if it was) and eigensolvers do not like this.
Zero row/column values for the boundary degrees of freedom of both matrices $[K]$ and $[M]$, and fix the diagonal values in $[M]$ to any finite number.
For eigenvalue problems I have (almost) exclusively used 1, while I have used the other methods (or equivalent) for solving linear systems of equations.
Questions
What is the most common method used for imposing Dirichlet boundary conditions?
Is there any other method that I have not mentioned?
This question has been partially answered in here.