Kernels for certain kinds of linear transformations

Every associative algebra has an associated Lie algebra. For a matrix algebra, there is a linear transformation associated with this Lie product by fixing one variable. The well-known dimensional formula of the kernel is due to Frobenius. Subsequently, Gracia obtained the dimensional formulas of the kernels of the second and third powers of the transformation. We fix two matrices and obtain a linear transformation. By using techniques from elementary linear algebra, together with the image spaces of the powers of the transformation, this paper provides an alternative approach to this problem. We obtain the dimensional formulas for kernels of each power of the transformation. Furthermore, the basis for kernels of powers of the transformation is described explicitly. In particular, we give the general solution for [A, . . . , [A, X]] = 0 for the k-fold Lie product with k > 1.