  In this paper, we calculate the Jordan decomposition (or say, the Jordan
canonical form) for a class of non-symmetric Ornstein-Uhlenbeck operators with
the drift coefficient matrix being a Jordan block and the diffusion coefficient
matrix being identity multiplying a constant. For the 2-dimensional case, we
present all the general eigenfunctions by the induction. For the 3-dimensional
case, we divide the calculating of the Jordan decomposition into several steps
(the key step is to do the canonical projection onto the homogeneous Hermite
polynomials, next we use the theory of systems of linear equations). As a
by-pass product, we get the geometric multiplicity of the eigenvalue of the
Ornstein-Uhlenbeck operator.
