(VI.E) Jordan Normal Form Set V = Cn and let T : V ! V be any linear transformation, with distinct eigenvalues s1,..., sm. In the last lecture we showed that V decomposes into stable eigenspaces for T : s s V = W1 ⊕ · · · ⊕ Wm = ker (T − s1I) ⊕ · · · ⊕ ker (T − smI). Let B = fB1,..., Bmg be a basis for V subordinate to this direct sum and set B = [T j ] , so that k Wk Bk [T]B = diagfB1,..., Bmg. Each Bk has only sk as eigenvalue. In the event that A = [T]eˆ is s diagonalizable, or equivalently ker (T − skI) = ker(T − skI) for all k , B is an eigenbasis and [T]B is a diagonal matrix diagf s1,..., s1 ;...; sm,..., sm g. | {z } | {z } d1=dim W1 dm=dim Wm Otherwise we must perform further surgery on the Bk ’s separately, in order to transform the blocks Bk (and so the entire matrix for T ) into the “simplest possible” form. The attentive reader will have noticed above that I have written T − skI in place of skI − T . This is a strategic move: when deal- ing with characteristic polynomials it is far more convenient to write det(lI − A) to produce a monic polynomial. On the other hand, as you’ll see now, it is better to work on the individual Wk with the nilpotent transformation T j − s I =: N . Wk k k Decomposition of the Stable Eigenspaces (Take 1). Let’s briefly omit subscripts and consider T : W ! W with one eigenvalue s , dim W = d , B a basis for W and [T]B = B. The operator N = T − sI 1 2 (VI.E) JORDAN NORMAL FORM d (with matrix [N]B = B − sId ) is nilpotent, and so fN(l) = l (since its only eigenvalue is 0 ). Since fN(l) must be the product of the invariant factors (of lI − N), the normal form of lI − N is quite limited: (r) (d) n f (lI − N) = diagf1, . , 1, lq ,..., lq g where1 q(r) + ... + q(d) = d. To express the corresponding rational canonical form for N we introduce the following notation: 0 s 1 B .. C q B 1 . C N := B C [q × q matrix]. s B .. .. C @ . A 1 s Then q(r) q(d) (1) R(N) = diag N0 ,..., N0 ; for instance, if 0 1 1 B C B 1 C B C n f (lI − N) = B 1 C B C B 2 C @ l A l3 then 0 1 0 B C B 1 0 C B C B C R(N) = B C . B 0 C B C B 1 0 C @ A 1 0 (Notice the apparent “gap” between the first and second 1’s.) 1once and for all: the numbers (r) ,..., (d) are not exponents, just superscripts. (VI.E) JORDAN NORMAL FORM 3 So for some change-of-basis matrix S we have −1 B − sId = [N]B = S R(N) S or −1 −1 B = S R(N) S + sId = S (R(N) + sId) S . Using (1) this means that if, say, S = SC!B then (r) (d) −1 q q [T]C = S BS = diag N0 ,..., N0 + sId q(r) q(d) = diag Ns ,..., Ns ; for example, 0 1 s B C B 1 s C B C B C n o B C = diag N 2, N 3 . B s C s s B C B 1 s C @ A 1 s q This is called a big Jordan block, and the “boxes” Ns are little Jordan blocks. They have very easy characteristic polynomials, namely (l − s)q (after all, T : W ! W has only eigenvalue s ). The little blocks correspond to a decomposition of W into N -cyclic subspaces W = W(r) ⊕ · · · ⊕ W(d) as in the rational canonical structure theorem (§VI.C). C is a basis subordinate to this decomposition, consisting of a choice of “cyclic vector” for each W(i) and its successive images under N ; more on this later. Going back to a transformation T : V ! V with multiple eigen- values, what we are aiming at is a further decomposition of each of s the Wk = ker (T − skI) into Nk -cyclic subspaces: (r1) (d1) (rm) (dm) V = W1 ⊕ · · · ⊕ W1 ⊕ · · · ⊕ Wm ⊕ · · · ⊕ Wm . 4 (VI.E) JORDAN NORMAL FORM With respect to some special basis C subordinate to this entire direct sum decomposition2 the matrix of T will then be (r ) (d ) ( ) ( ) q 1 q 1 rm dm J ( ) = N 1 N 1 N qm N qm T diag s1 ,..., s1 ;...; sm ,..., sm , where each semicolon separates two big Jordan blocks3 (and each comma two little Jordan blocks). As it stands this looks formidable, and at least the basis C looks very hard to compute. Decomposition of Wk (Take 2). So in order to facilitate compu- tation of the entire Jordan decomposition A = SJ S−1 , it is useful to have an approach to the “nilpotent building blocks” Nk : Wk ! Wk that does not appeal to rational canonical form. DEFINITION 1. The height h of a nilpotent transformation N : W ! W is defined by Nh = 0, Nh−1 6= 0. (The minimal polynomial of such a transformation is simply lh .) One may also define the (N-)height h(w~ ) of w~ 2 W by Nh(w~ )w~ = 0, Nh(w~ )−1w~ 6= 0 . Clearly the height of N is just the supremum of the (N-)heights of vectors 2 W . (The height of a subspace of W, similarly, just means the supremum of heights of vectors in that subspace.) h0 Notation: We shall write ZN (w~ ) for the N -cyclic subspace of W generated by w~ (w~ of height h0 ); that is, spanfw~ , Nw~ ,..., Nh0−1w~ g. CLAIM 2. Given N : W ! W nilpotent of height h , with h−1 h−1 fN w~ 1,..., N w~ tg a basis for im(Nh−1). Then we have a decomposition h h (t+1) (s) W = Z (w~ 1) ⊕ · · · ⊕ Z (w~ t) ⊕ W ⊕ · · · ⊕ W . N N | {z } cyclic of height ≤h−1 2but this is not enough; “special” means the basis also has “cyclic” properties (to be described below) 3there is one for each eigenvalue (VI.E) JORDAN NORMAL FORM 5 PROOF (BY INDUCTION ON h). h = 1 : W = kerN . Take any basis for W ; the spans of the individual basis vectors give the desired decomposition. inductive step : The following picture of W is useful, where N acts by sending you up one level, the top row is kerN , and the height of a vector is its “distance” from the top. h−1 im(N ) (top=0) h−1 h−1 N w N w ker(N) N 1 t N N w w 1 t In order to apply the inductive hypothesis (= the Claim for height h−1 h − 1 ), we consider U = kerN ⊆ W . Now N j U has height h − 1: h−2 im(N| U ) h−1 h−1 h−2 h−2 N w........ N w N u N u 1 t t+1 r Nw 1 ..........Nwt ut+1 ...........u r Set f~u1,...,~utg = fNw~ 1,..., Nw~ tg ⊆ U , and complete h−2 h−2 h−1 h−1 fN ~u1,..., N ~utg = fN w~ 1,..., N w~ tg 6 (VI.E) JORDAN NORMAL FORM h−2 h−2 h−2 to a basis for imf(N j U) g , by adding some fN ~ut+1,..., N ~urg as shown. By the inductive hypothesis, h−1 h−1 h−1 h−1 U = ZN (~u1) ⊕ · · · ⊕ ZN (~ut) ⊕ ZN (~ut+1) ⊕ · · · ⊕ ZN (~ur) ⊕ U(r+1) ⊕ · · · ⊕ U(s) . | {z } N-cyclic of height≤h−2 The picture is now h−1 h−1 h−2 h−2 N w........ N w N u t+1 N ur 2 1 2 t NwNw Nu Nu 1 t t+1 r Nw ..........Nw u ...........u 1 t t+1 r guaranteed by inductive hypothesis — U is the direct sum of the columns (= cyclic subspaces under the action of N ). So why not tack w~ 1,..., w~ t on to the bottom of the first t columns and obtain a decomposition . Nw Nw 1 t w w 1 t as promised? h−1 h−1 Here’s how to do this rigorously: since N w~ 1,..., N w~ t are h−1 a basis for im(N ) , no nontrivial linear combination of w~ 1,..., w~ t can be in ker(Nh−1) = U . Therefore (2) spanfw~ 1,..., w~ tg \ U = f0g, which implies that dim(spanfw~ 1,..., w~ tg) + dim U = dim(spanfw~ 1 ... w~ tg + U). (VI.E) JORDAN NORMAL FORM 7 Rank+Nullity applied to Nh−1 =) dim W = dim(spanfw~ 1,..., w~ tg + U) =) W = spanfw~ 1,..., w~ tg + U. Together with (2) this =) W = spanfw~ 1,..., w~ tg ⊕ U, which by our decomposition of U h−1 h−1 = spanfw~ 1,..., w~ tg ⊕ ZN (~u1) ⊕ · · · ⊕ ZN (~ut) h−1 h−1 (r+1) (s) ⊕ZN (~ut+1) ⊕ ... ⊕ ZN (~ur) ⊕ U ⊕ ... ⊕ U . Since f~u1,...,~utg was just fNw~ 1,..., Nw~ tg , the direct sum in the parentheses is just h h ZN(w~ 1) ⊕ · · · ⊕ ZN(w~ t) and we are done. The upshot of the Claim just proved is this (weaker) PROPOSITION 3. N : W ! W nilpotent =) (1) (s) h1 hs W = W ⊕ · · · ⊕ W = ZN (w~ 1) ⊕ · · · ⊕ ZN (w~ s); that is, W decomposes into a direct sum of N -cyclic subspaces of various heights.4 So if we take h1−1 hs−1 C = fw~ 1, Nw~ 1,..., N w~ 1;...; w~ s, Nw~ s,..., N w~ sg, then n o h1 hs [N]C = diag N0 ,..., N0 .