GENERALIZED COMMUTATORS AND THE MOORE-PENROSE INVERSE

This work studies the kernel of a linear operator associated with the generalized k-fold commutator. Given a set u= {A(1), ..., A(k)} of real n x n matrices, the commutator is denoted by [A(1)vertical bar...vertical bar A(k)]. For a fixed set of matrices u we introduce a multilinear skew-symmetric linear operator T-u (X) = T(A(1), ..., A(k)) [X] = [A(1)vertical bar...vertical bar A(k)vertical bar X]. For fixed n and k >= 2n - 1, T-u 0 by the Amitsur-Levitski Theorem [2], which motivated this work. The matrix representation M of the linear transformation T is called the k-commutator matrix. M has interesting properties, e.g., it is a commutator; for k odd, there is a permutation of the rows of M that makes it skew-symmetric. For both k and n odd, a provocative matrix S appears in the kernel of T. By using the Moore-Penrose inverse and introducing a conjecture about the rank of M, the entries of S are shown to be quotients of polynomials in the entries of the matrices in u. One case of the conjecture has been recently proven by Brassil. The Moore-Penrose inverse provides a full rank decomposition of M.