  The adjoint of a matrix in the Lie algebra associated with a matrix algebra
is a fundamental operator, which can be generalized to a more general operator
$\varphi_{AB}: X\rightarrow AX-XB$ by two matrices $A$ and $B$. The kernel of
the operator is very well-known and it can be found in Gantmacher's book. The
formulas for the dimensions of the kernels of arbitrary powers of the operator
$\varphi_{AB}$ were given in terms of the Segre characteristics of these two
matrices by the second and third authors in this paper and their collaborators.
This paper provides an alternative approach to this problem via the Weyr
characteristic in a more essential method. We obtain formulas for the
dimensions of the kernels of arbitrary powers of the operator in terms of the
Weyr characteristics. Furthermore, the basis for kernel of each power of the
operator is described explicitly. As a consequence, for arbitrary square
matrices $A$ and $B$ over an algebraically closed field, the dimension of the
kernel of each power of the operator $\varphi_{A-\lambda I,B}$ for eigenvalues
$\lambda$ of $\varphi_{AB}$ can be viewed as a similarity invariant of the
operator $\varphi_{AB}$, so we characterise the operator within similarity,
which should be of interest to a number of people (including physicists).
