The reduction of a matrix which has several roots is treated. For example, when a characteristic root of a matrix A of order n is known, it is possible, by any of sev eral known methods, to replace A by a matrix B, n-1 of whose roots are the yet unknown roots of A, and such that either B is of order n-1, or else its remaining root is zero. It is shown that the vector iterates approach an invariant subspace belonging to these roots, and even when the roots are close, the corresponding invariant subspace is more stably separable from the complementary invariant subspace than are the principal vectors from one another within the subspace. By repeated application of the equations listed, the invariant subspaces can be peeled off and the problem reduced, at each step, to one of lower order. (N.W.R.)