Arch. Math., Vol. 50, 113-117 (1988) 0003-889X/88/5002-0113 $2.50/0 9 1988 Birkhfiuser Verlag, Basel Jordan normal form projections By NEE YUEN LAM, ANDREW RANICKI a n d LARRY SMITH Let K be a field, and let T: V ~ V be an endomorphism of a finite-dimensional vector space V over K such that K contains the roots 21, 22, ..., )-,, of the characteristic polyno- mial 7~(t) = det (tI - T)= (t - )ol) "~ ( t - 22) ..... (t - 2m)nm . V is a direct sum of T-invariant subspaces Vj = ker (T - 2jI)"J, one for each eigenvalue 2j, such that T - 2jI: Vk ~ Vk is nilpotent forj = k and an automorphism forj 4= k. We obtain in this paper an explicit formula for the projection pj (T) = pj (T)Z: V ~ V onto the subspace V~, as a polynomial in T. A near-projection in a ring A is an element p ~ A such that q = p (1 - p) ~ A is nilpotent, that is q" = 0 for some exponent n > 1. We refer to Lfick and Ranicki [3] for the general theory of near-projections, and for the construction of the unique projection p~, = (p,o) 2 e A such that p~ - p is nilpotent and pp~ = p~p, namely po, = (p" + (1 -p)")-lp, = p + (1/2)(2p - 1)((i - 4 q)-1/2 __ 1) =p + (2p -- l)(q + 3q z + 10q 3 + 35q 4 + 126@ + 462q 6 + 1716q 7 + 6435q s + 24310q 9 + 92378 q 1~ + 352716q 11 + 1352078q lz + "" cA. (In the special case when A is of characteristic 2 this simplifies to Po, = P + q + q2 + q4 + q8 + q16 + ... E A). If R is any ring and p: V ~ V is a near-projection in the endomorphism ring of an R-module V then P,o: V --* V is the projection onto the p-invariant submodule P = im (p~) = im (p") = ker ((1 - p)") ~ V, such that V = P @ Q with Q the p-invariant submodule Q = im (1 - pj = im ((1 - p)") = ker (p") ~ V. p: P --* P is an isomorphism and p: Q ~ Q is nilpotent. An endomorphism p: V ~ V of a finite-dimensional vector space V over a field K is a near-projection if and only if the characteristic polynomial is of the type z(t) = t" (1 - t)", in which case (p (1 -p))~a~(m,,) =0:V~V Archiv der Mathematik 50 8