  We characterize the eigenvalues and eigenvectors of a class of complex valued
tridiagonal $n$ by $n$ matrices subject to arbitrary boundary conditions, i.e.
with arbitrary elements on the first and last rows of the matrix. %By boundary
conditions, we mean the first and last row of the matrix. For large $n$, we
show there are up to $4$ eigenvalues, the so-called \emph{special eigenvalues},
whose behavior depends sensitively on the boundary conditions. The other
eigenvalues, the so-called \emph{regular eigenvalues} vary very little as
function of the boundary conditions. For large $n$, we determine the regular
eigenvalues up to ${\cal O}(n^{-2})$, and the special eigenvalues up to ${\cal
O}(\kappa^n)$, for some $\kappa\in (0,1)$. The components of the eigenvectors
are determined up to ${\cal O}(n^{-1})$.
  The matrices we study have important applications throughout the sciences.
Among the most common ones are arrays of linear dynamical systems with nearest
neighbor coupling, and discretizations of second order linear partial
differential equations. In both cases, we give examples where specific choices
of boundary conditions substantially influence leading eigenvalues, and
therefore the global dynamics of the system.
