On the centralizer of the centralizer of a matrix Original Research Article

Let Mn(F) be the algebra of all n × n matrices over an arbitrary field F, and for S, T subset of Mn(F), let imageT(S) = B set membership, variant T: AB = BA for all A set membership, variant S. For each A set membership, variant Mn(F), it is well known that imageMn(F)(A)) = F[A], the algebra of polynomials in A over F. We determine imageMn(F)(imageGL(n, F)(A)). It turns out that imageMn(F)(imageGL(n, F)(A)) = F[A] when F > 2. When vbF VB = 2, imageMn(F)(imageGL(n, F)(A)) = F[A] unless A has both 0 and 1 as eigenvalues and the elementary divisors of greatest degree corresponding to 0 and 1 are not repeated in the list of elementary divisors. In this exceptional case, imageMn(F)(imageGL(n, F)(A)) = F[A] circled plus R, where R is an explicitly described two dimensional subspace of Mn(F). We also determine imageGL(n, F)(imageGL(n, F)(A)).