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8
EIGENVALUE ANALYSIS
The behavior of the eigenvalues of the matrices introduced in chapter seven is interesting
for two reasons. The first is related to the application of iterative methods for the
solution of the corresponding systems of equations. The second is to examine the
approximation of eigenvalue problems for differential operators, which arise in various
applications in mathematical physics.
8.1 Eigenvalues of first derivative operators
All the eigenvalues of the derivative matrices presented in sections 7.1 and 7.2, are equal
to zero. Actually, if there existed one eigenvalue different from zero, a corresponding
polynomial eigenfunction would not vanish after repeated applications of the derivative
operator. This is in contradiction to one of the fundamental theorems of calculus.
The situation changes when we introduce auxiliary constraints, such as boundary con-
ditions. Now, since the matrix is required to be non singular, all the corresponding
eigenvalues must be different from zero.
Let us begin by analyzing the eigenvalues of the matrix associated with problem
(7.4.1), relative to the Jacobi case with n ≥ 1, α > −1 and β > −1. More precisely,