  Motivated by the challenge of seeking a rigorous foundation for the
bulk-boundary correspondence for free fermions, we introduce an algorithm for
determining exactly the spectrum and a generalized-eigenvector basis of a class
of banded block quasi-Toeplitz matrices that we call corner-modified. Corner
modifications of otherwise arbitrary banded block-Toeplitz matrices capture the
effect of boundary conditions and the associated breakdown of translational
invariance. Our algorithm leverages the interplay between a non-standard,
projector-based method of kernel determination (physically, a bulk-boundary
separation) and families of linear representations of the algebra of matrix
Laurent polynomials. Thanks to the fact that these representations act on
infinite-dimensional carrier spaces in which translation symmetry is restored,
it becomes possible to determine the eigensystem of an auxiliary projected
block-Laurent matrix. This results in an analytic eigenvector Ansatz,
independent of the system size, which we prove is guaranteed to contain the
full solution of the original finite-dimensional problem. The actual solution
is then obtained by imposing compatibility with a boundary matrix, also
independent of system size. As an application, we show analytically that
eigenvectors of short-ranged fermionic tight-binding models may display
power-law corrections to exponential decay, and demonstrate the phenomenon for
the paradigmatic Majorana chain of Kitaev.
