A Tridiagonal Matrix

 = αI +βT, where T is defined by the preceding formula. This matrix arises in many applications, such as n coupled harmonic oscillators and solving the Laplace equation numerically. Clearly M and T have the same eigenvectors and their respective eigenvalues are related by μ = α+βλ . Thus, to understand M it is sufficient to work with the simpler matrix T . Eigenvalues and Eigenvectors of T Usually one first finds the eigenvalues and then the eigenvectors of a matrix. For T , it is a bit simpler first to find the eigenvectors. Let λ be an eigenvalue (necessarily real) and V = (v1,v2, . . . ,vn) be a corresponding eigenvector. It will be convenient to write λ = 2c . Then 0 = (T −λI)V = 