150 CHAPTER 6. DIRECT SUMS AND BLOCK DIAGONAL MATRICES

with the size of the diagonal blocks decreasing and Jm (µk) having ones down thesuper diagonal and µk down the diagonal. Hint: Explain why, for I the identity onVk the matrix of µkI with respect to any basis is just the diagonal matrix having µkdown the diagonal. Thus the matrix of A restricted to Vk relative to the basis β kwill be of the desired form. Note that on Vk,A = Nk + µkI. This yields the Jordancanonical form. Another argument based on rings and modules will be introduced inthe following chapter to obtain both the rational and Jordan form.