! } Extensions of Jordan Bases for Invariant Subspaces of a Matrix

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1 1 ( i \! } Extensins f Jrdan Bases fr Invariant Subspaces f a Matrix Rafael Bru* Departament Matematica Aplicada Univ. Plitecnica de Valencia Valencia, Spain Leiba Rdman t Department f Mathematics Cllege f William and Mary Williamsburg, Virginia 23187~8795 and Hans Schneider * Department f Mathematics University f Wiscnsin Madisn, Wiscnsin Submitted by Daniel Hershkwitz ABSTRACT A characterizatin is btained fr the matrices A with the prperty that every (sme) Jrdan basis f every A-invariant subspace can be extended t a Jrdan basis f A. These results are based n a criterin fr a Jrdan basis f an invariant subspace t be extendable t a Jrdan basis f the whle space. The criterin invlves tw cncepts: the cnstancy prperty and the depth prperty. "'Supprted by Cnselleria de Cultura Educacin y Ciencia de la Generalitat Valenciana. and by D.C.1. c.y.t. f Ministeri de Educacin y Ciencia, Spain. tpartially supprted by NSF grant DMS *Partially supprted by NSF grants DMS , DMS , ECS LINEAR ALGEBRA AND ITS APPliCATIONS 150: (1991) Elsevier Science Publishing C., Inc., Avenue f the Americas, New Yrk, NY /91/$3.50

2 210 RAPHAEL BRU, LEIBA RODMAN, AND HANS SCHNEIDER 1. INTRODUCTION Let A be an n X n cmplex matrix cnsidered as a linear transfnnatin en -., en. A chain (fr A) is a set f nnzer vectrs {u,(a - AI)u,..., (A - Al)k -lu} (1.1) such that (A - AnkU = O. The cmplex number A is necessarily an eigenvalue f A and (A - AI)k-Iu is an eigenvectr. A Jrdan basis fr an invariant subspace W is a basis fr W which is the unin f chains. A Jrdan basis f en will be called a Jrdan basis fr A. That is, a Jrdan basis fr A is a basis f the frm where U j E en and (A - AI)klUi = O. The existence f a Jrdan basis fr any n X n matrix A is well knwn and fllws frm the existence f the Jrdan nnnal fnn f A. Given a Jrdan basis (1.2), certain A-invariant subspaces are seen immediately. Namely, fr any chice f integers m, (i = 1,..., t) such that 0 ~ mj ~ k j the subspace (1.3) is A-invariant. i.e., Ax EM fr every x EM [the equality mj = k, fr sme i is interpreted as the indicatin that i is missing in the frmula 0.3)]. The A-invariant subspaces that arise in this way, starting with any Jrdan basis, are called marked in [2]. Equivalently, an A-invariant subspace M is caned marked if there is a Jrdan basis fr the restrictin AIM: M -., M which can be extended (by adjining t it new vectrs) t a Jrdan basis fr A in en. Generally, nt every A-invariant subspace is marked (an example is given in [2]). The existence f nnmarked invariant subspaces is smetimes verlked in linear algebra texts. In this paper we characterize thse matrices A fr which every invariant subspace is marked. We als characterize the matrices A with a strnger prperty, namely, that every A-invariant subspace is strngly marked. Let us define this ntin: an A-invariant subspace M is strngly marked if every Jrdan basis f M can be extended (by adjining

3 EXTENSIONS OF JORDAN BASES 211 new vectrs) t a Jrdan basis fr A in en. These ntins call ur attentin t a mre general questin: when can a given Jrdan basis fr an' A-invariant subspace be extended t a Jrdan basis fr the whle space en? We slve this prblem in Sectin 2 in tenns f the height and depth f vectrs and related prperties. Anther characterizatin Gn different tenns} f this extendability prperty is given in [1]. These results are used in subsequent sectins t characterize marked and strngly marked subspaces. This characterizatin ges as fllws. (The multiplicities f a matrix A crrespnding t its eigenvalue A are simply the sizes f the Jrdan blcks with the eigenvalue A in the Jrdan nnnal frm f A.) THEOREM 1.1. Let A be an n X n matrix. Then every A-invariant subspace is marked if and nly if fr every eigenvalue A 0 f A the difference between the biggest and the smallest multiplicity f A crrespnding t A des nt exceed 1. THEOREM 1.2. Let A be an n X n matrix. Then every A-invariant subspace is strngly marked if and nly if fr every eigenvalue A f A all multiplicities f A crrespnding t A 0 are equal. T illustrate these results cnsider the fllwing example. Let A=[! 1 Accrding t Therems 1.1 and 1.2, every A-invariant subspace is marked, but there are A-invariant subspaces which are nt strngly marked. Fr example, K(A) is nt strngly marked. Here and elsewhere in this paper K(A) stands fr the kernel (null space) f the matrix A. Indeed, a Jrdan basis fr A,K(A) given by (a},0,/32,0,1'l)t,(a2,o,/3z,0,'yz)t,(a3,0,/33,0,'y3)t (here a j, f3 j, 1'j, E e) can be extended t a Jrdan basis fr A if and nly if there are tw zers amng the nulllbers 'Y l' l' 2' 'Y 3' An easy (but smewhat tedius) analysis shws that the fllwing is a L'Omplete list f all A-invariant subspaces which are nt strngly marked: K(A); all 2-dimensinal A invariant subspaces spanned by eigenvectrs, with the exceptin f Span{(l, 0, 0, 0, O)T, (0, 0,1,0, O)T); all 4-dimensinal A-invariant subspaces cntaining K(A). 1!I

4 212 RAPHAEL BRU. LEIBA RODMAN, AND HANS SCHNEIDER As a crnary we recver the fllwing result frm [2] (Therem 2.9.2). In fact the cnclusin f Therem f [2] is weaker in the sense that nly the marked prperty f every A-invariant subspace is asserted there. COROLLARY 1.2. l t A be an n X n matrix such that fr every eigenvalue A f A at least ne f the fllwing hlds: (a) the gemetric multiplicity (i.e., the dimensin f K(A - AI) is equal t the algebraic multiplicity; (b) dim K(A - AI) = 1. Then every A-invariant subspace is strngly marked. The prfs f Therems 1.1 and 1.2 will be given in Sectins 3 and 4, respectively. We cnclude the intrductin by remarking that it is sufficient t prve Therems 1.1 and 1.2 (and Therem 2.1 stated belw) fr the case when A has a single eigenvalue A (withut lss f generality it can be assumed that A = 0). This fllws readily frm the well-knwn fact that every A-invariant subspace M can be written as where A l'..., A r are all the distinct eigenvalues f A and is the rt subspace f A crrespnding t A j' Thus, it will be assumed in Sectins 2, 3, and 4 that A is nilptent: An =. 2. HEIGHT AND DEPTH Let A be an n X n nilptent cmplex matrix. Fr a given x E en let the height f x [ntatin: ht(x)] be the minimal nnnegative integer k such that Akx = 0 (as usual, we assume AO = 1; thus zer is the nly vectr f height zer). Fr x :1= 0, the depth f x [ntatin: dpth(x)] is by definitin the maximal nnnegative integer k such that

5 EXTENSIONS OF JORDAN BASES 213 x = Ak y fr sme y. Nte the fllwing easily verified prperties: Fr cmplex numbers a I'..., a Ii' and vectrs x I'..., x II we have ht(.t aixi) ~ max{ht(xj: i = 1,..., s},,= 1 (i) dpth (,t aix t ] ~ min{ dpth( Xi): i = 1,..., s}, 1=1 (ii) and the strict inequality dpth( x) "* dpth( y) => dpth( x + y) = min{ dpth( x), dpth( y)}. (iii) prvided all vectrs in (ij) and (iii) are nnzer. Als, fr 0"* u E e", we have ht( Au) = ht( u) - I, (iv) dpth( Au) > dpth( u), ( v) prvided that Au "* O. We address the questin when a given Jrdan basis B (1.2) fr W can be extended t a Jrdan basis fr the whle space en, i.e., when there is a Jrdan basis T in en such that B ~ T (as sets f vectrs). The answer is based n tw ntins that we call the cnstancy prperty and the depth prperty. We say that a nnzer vectr x has the cnstancy prperly (CP) if either Ax = 0 r Ax =1= 0 and dpth( Ax) = dpth( x) + l. A set S f nnzer vectrs is said t have' the CP if every vectr in S has the CP In particular, the ntin f the cnstancy prperty can be applied t a chain S = {x, Ax,..., Ak -1 x}; thus, this chain has CP if and nly if dpth(ai-lx) = ' dpth( x) + i -1, i=i,...,k. (2.1)

6 214 RAPHAEL BRU. LEIBA RODMAN, AND HANS SCHNEIDER As by (iv) ht{aix) = k - i (0 ~ i ~ k - 1), these equalities can be rewritten in the fnn dpth( x) + ht( x) = dpth( A'x) + ht(aix), i = 0,..., k - 1. ( 2.2) Als, if dpth(ak-1x) = k -1, then necessarily dpth(x) = and (2.2) hlds, and thus the chain {x, Ax,..., Ale-IX}' has the CP In what fllws we use the ntatin (q) fr the set {I,..., q}. We say that a linearly independent set f vectrs {Xi: i E (q)} has the DP (the depth prperty) if w = EiE(q)a,x i, w F 0, implies that dpth( w) = min {dpth ( Xi) : i e (q) and a i :;: O}. (2.3) The tw prperties CP and DP d nt imply each ther, as examples will presently shw. First, nte that every chain {x, Ax,..., Ale-IX} is linearly independent, by a standard argument. It fllws frm (iii) and (v) that every chain has the DP. An example f a chain withut the CP (but with DP) is furnished by {u, Au} where ~l and u = (0, 1, 0, l)t. The fllwing example shws a linearly independent set f chains withut the DP. Let u = (1,0, 1)T, and v = (0,0, 1)T. Then each f the (singletn) chains {u} and (v) has the CP, the set {u, v} is linearly independent, but (u, v} des nt have the DP. Indeed, dpth(u) = dpth(v) = 0, but fr a nnzer vectr w = au + f3v we have dpth(w) = 0 if a + f3 =F 0 and dpth(w) = 1 if a + f3 =. The main result f this paper is the fllwing (which hlds withut the assumptin that A is nilptent, thugh the prf is given nly fr nilptent A; see the end f Sectin 1). 1 THEOREM 2.1. Let A be a cmplex n X n 1TUltriX. Let B be a Jrdan basis fr an A-invariant subspace W. Then B can be extended t a Jrdan basis fr

7 EXTENSIONS OF JORDAN BASES 215 A in en if and nly if B has the CP and the DP. Prf. " If": Let be a Jrdan basis fr cn. It is enugh t prve that C has the CP and the DP, fr then any subset f C has the CP and the DP. Let w be a nnzer vectr in cn. Then w can be written uniquely as t k;-l w = E E aijaju ;, j = 1 j =0 aijec, j=o,...,k j -l, i=i,...,t. Then, fr p ~ 0, A,IW has the unique representatin I k;-l AflW = E E ai.j_pajui' ;=1 j=o j=o,...,k;-i, i=i,...,t, where aij = whenever j < 0, i = 1,..., t. If Aw =1= 0, then it fllws easily that In particular, dpth( w) = min{ j: at least ne f a ij, i = 1,..., t, is nnzer). (2.4) j=o,...,k j -l, i=i,...,t. (2.5) Thus, by (2.1), C has the CP, and, by applying (2.5) t (2.4) we see that C has the DP. Hence if B is a subset f C, then B has the CP and the DP. "Only if": We suppse that W =1= C", fr theiwise f curse there is nthing t prve. We cnsider tw cases. In each case we cnstruct a subspace W' which prper]y cntains W and a Jrdan basis B' ~ B fr W' such that B' has the CP and the DP. We say that a chain {u, Au,..., Ak-1u}' is maximal if it is nt cntained (set theretical1y) in a larger chain; in ther wrds, a chain {u, Au,..., Ak - lu} is maximal if dpth(u) = O. Case 1: Sme chain f B is nt maximal. Suppse that S = {u,..., Ah-1u} is a chain f B, and that dpth(u) = d> O. Let y E en satisfy Ady = U.

8 216 RAPHAEL BRU, LEIBA RODMAN, AND HANS SCHNEIDER Then dpth(y)=o. Let S'={y,...,Ad+h-1y). We let B' cnsist f the chains f B with S replaced by S', and we let W' = span(b'). Claim 1.1. B' is linearly independent. Otherwise, there exists a nntrivial linear relatin n B', and since this cannt be a nntrivial linear relatin n B, it must invlve an element f frm Ary, where r < d. We chse the minimal such r. Multiplying this linear relatin by A d -,., we btain a linear relatin n the elements f B, which is nntrivial, since it invlves Ady = U. But this is impssible, since B is linearly independent. Claim 1.1. The chain 5' = {y,..., Ad +h - 1 y) has the CP. Otherwise, by (2.1), dpth(ad+h-1)y) > d + h -I, and there is a y' E en such that But then dpth(ah-1u) -dpth(u) ~ d + h - d = h, which is impssible by (2.1), since S has the CP Hence dpth(al1+h-1y) = d + h -1, and hence S' has the CP Claim 1.3. B' has the DP. Recall that every chain has the DP. Suppse B' des nt. Since Band S' have the DP, it is easily shwn using (iii) that there exists awe W and an x = E 'Y.yA~y, se (r... d -I) 'Yy E C, 'Yr '* 0, (2.6) where 0 ~ r < d, such that w"* 0, x * 0, dpth( w) = dpth( x) = r, (2.7) and, fr v = w + x, dpth( v) > r. (2.8) We then btain dpth ( Ad - r V ) ~ dpth ( v) + d - r > d. (2.9) But this is impssible, fr Ad-rv is a linear cmbinatin f nnzer elements

9 EXTENSIONS OF JORDAN BASES 217 f B ne f which is Ad y = u, and dpth(u)= d. This prves the claim, and cmpletes the prf f case I. Case II: Every chain f B is maximal. Claim II.l. There exists v E cn, v$. W, with ht(v) = 1 (i.e., v is an eigenvectr f A). Let u E cn, U $. W. Let ht(u) = h. If An-1u $. W, the claim is true. Otherwise there exists a least r, 0 < r < h - 1, such that A r u E W. Thus A r u is a linear cmbinatin f B, and, since B has the DP, it is a linear cmbinatin f elements f B whse depth is at least 1. Thus there exists a we W such that Aw = Aru. Let x = w - Ar-1u. Then x$. Wand Ax = 0, which prves the claim. We nw chse a chain S = {u..., Ali - 1 u} f maximal length such that Ah-1u = v is nt in W. Let 8' = BUS. Then it is easy t prve that W n span(s) = 0, and it fllws that B' is a basis fr W' = W E9 span(s). Claim II. 2. The chain S has the CP Since S is a maximal chain beginning at u, clearly dpth(u) = O. Suppse S des nt have the CP Then, by (2.1), dpth(a,,-iu»h-l. Hence there is a wec" such that Ahw=Ah-lu. Thus the chain {w,..., AIi-Iu} has greater length than S, cntrary t the assumptin that S is a maximal chain whse last element is nt in W. Claim II. 3. B' has the DP. Suppse B' des nt have the DP. Since Band the chain S have the DP, there must exist such that v=w+x, WE W, x E span(s), dpth( v) > min{ dpth( w ),dpth( x)}. (2.10) By OiO, we then have when 0 ~ d < h. By (2.11), we have dpth( w) = dpth(x) = d, say, (2.11) x = E 'Yr Aru, 'Yr E C, I'd * O. r E (d... h-l) By (2.10), there is a z E C" such that A'l+ lz = V. Then Ah - A h - d A h - d Ah- d Ah - d Ah- 1 Z - V - U? x - W 'Yd U.

10 218 RAPHAEL BRU, LEIBA RODMAN. AND HANS SCHNEIDER Since Ah-d-1w E Wand 'YdAh-IU::l= 0, it fllws that there is a chain f length h + 1 which ends utside W, cntrary t ur assumptin n S. This prves ur claim, and cmpletes the prf f case II. Thus in either case, we have cnstructed an invariant subspace W' with dim(w') > dim(w) and a Jrdan basis B' d B with the DP such that B' has the CP By repeating this argument we btain a Jrdan basis fr en, which is an extensin f B. Anther necessary cnditin fr extendability f B t Jrdan basis if en can be given in tenns f multiplicities, as fllws. We write the list f all multiplicities (including repetitins, if necessary) in a nn increasing rder: Al ~... ~ A q A sequence f psitive integers f31'"'' f3 p will be caned a sublist f multiplicities if p ~ q and there is a ne-t-ne map ~: {l,..., p} ~ {l,...,q} such that (3i = A W ) fr i = l,...,p. By the index f a chain S={x,...,Ak-Ix}, dented ind(s), we mean dpth(ak-1x)+1. By (2.4), it is easy t see that if a Jrdan basis B f W. is extendable t a Jrdan basis f en, then the numbers ind(si)' where SI'"'' Sr are the chains in B, fnn a sublist f multiplicities. The fllwing example shws that a Jrdan basis B with the CP and fr which ind(s), i E (r), fnn a sublist f multiplicities need nt be extendable t a Jrdan basis f en, EXAMPLE 2.1. Let A=[~ n Let u = (1,0, 1,0)T, V = (0,0, 1,0)T, Then (u, v} frms a Jrdan basis B f the subspace W= span{(i,o,o,o)t,(o,o,l,o)t}. By Therem 2.1 the basis B cannt be extended t a Jrdan basis in e 4, since dpth(u) = dpth(v) = 0, while dpth(u - v) = 1. Hwever, the basis B has the CP and {I, I} is a sublist f multiplicities, 3. PROOF OF THEOREM 1.1 Let A be an n X n nilptent matrix. We start with the fllwing: PROPOSITION 3.1. Suppse that every A-invariant subspace is marked. Then the lengths f any tw maximal chains (in a Jrdan basis f A) differ by less than tw.

11 EXTENSIONS OF JORDAN BASES 219 Prf. Arguing by cntradictin, assume that is a maximal chain in a Jrdan basis f A, and u, Au,..., A r - 3 U (3.1) v, Av,..., AT-IV is a nt necessarily maximal chain in the same Jrdan basis f A. Put z = u + Av. Nte that A r - 2 z = A r - lv. Further nte that dpth(z) = 0 [indeed, if Ay = z fr sme y, then u = A(y - v), which is a cntradictin with the maximality in (3.1)]. We have ht(a r - 2 z) = 1, dpth(a r - 2 z) ~ r -1, ht(z) = r -1, and dpth(z) = 0; s the chain z, Az,..., A r 2 - z des nt have the CP and hence by Therem 2.1 cannt be extended t a Jrdan basis fr en. Observe that every Jrdan basis fr span{z..., AT-2Z} has the frm where r -2 W = E ajajz, j=o { w, Aw,..., AT - 2 w}, We see that dpth(z) = 0 and dpth(a r - 2 w) = dpth(aa T 2 - w) = dpth(a A,.-2z ) ~ r -1. Thus, the chain {AJw};:J des nt have the CP, and by Therem 2.1 it cannt be extended t a Jrdan basis fr A. Therefre, the A-invariant subspace span{z,..., A r - 2 z} is nt marked. PROPOSITION 3.2. Let A be an n X n nilptent matrix with sizes f all Jrdan blcks equal t q r q - L Then all chains have the CP Prf. Let w be a vectr in e'l such that Aw =1= O. We shall shw that ht( w) + dpth( w) = ht( Aw) + dpth( Aw ). (3.2) It fllws frm ur assumptins that we may write w=u +v, where u and v are linear cmbinatins f vectrs in Jrdan chains f lengths,

12 220 RAPHAEL BRU, LEIBA RODMAN, AND HANS SCHNEIDER respectively. q and q -1. Suppse that ht( u) = h, ht( v) = k. Then 0 ~ h ~ q, 0 ~ k ~ q -1. If h = 0 (i.e. u = 0) r k = 0 (i.e. v = 0), then we are basically in the situatin [as far as (3.2) is cncerned] when all the multiplicities f A are equal. But in this case (3.2) fllws easily [see als the equivalence (1) ~ (2) in rherem 1.2' f Sectin 4]. S suppse that h, k ~ l. Nte that we cannt have h = k = 1, fr then Aw = 0, cntrary t assumptin. S either h > 1 r k > 1. It is easily checked that ht(w) = max{h.k}, dpth( w) = min{ q - h, q -1- k}, ht(aw) =, max{h -1, k -I}. dpth(aw) = min{q - h + 1,q - k}. If h > k, it fllws that while if h ~ k ht( w) + dpth( w) = ht( Aw ) = dpth( Aw) = q, ht(w)+dpth(w) =ht(aw)+dpth(aw) =q-l. In either case, (3.2) hlds and the prpsitin fllws. PROPOSITION 3.3. Let A be an n X n nilptent matrix with sizes f all Jrdan blcks equal t q r q -1. Let M be an A-invariant subspace f ell, and let B be a Jrdan basis fr M with a maximal number f eigenvectrs f depth q -1. Then B has the DP. Prf. Let B be the Jrdan basis

13 EXTENSIONS OF JORDAN BASES 221 and let k = max { k i : i = 1,..., t}. Let B h, h = 1,..., k, be the subset f B cnsisting f vectrs f height h r less, viz.. We shall prve by inductin that B h, h = 1,..,k, has the DP. We first cnsider B 1 In view f ur assumptins n multiplicities, each vectr in B 1 has depth q - 2 r q - 1 (since it is an eigenvectr f A). Cnsider the linear cmbinatin f B}: p 0=1= w = E ajakj-lg j + {3.3} j= I where we may assume that i=l,...,p, i=p+l,...,t. Nw suppse that B1 des nt have the DP. Then we may find a vectr w f fnn (3.3) such that at least ne f the cefficients ai' 1 ~ i ~ p, is nnzer and dpth( w) = q - 1. But then I> V = E ajaki-lg i i= 1 (3.4) als satisfies dpth( v) = q - 1 by (iii). Let s, 1 ~ 8 ~ p, be an index fr which as =1= 0 in (3.4) and such that k ~ is minimal amng k j fr which a i =1= 0 in (3.4). Suppse, withut lss f generality, that (Xi =1= 0, i = 1,..., s and (Xi = 0, i = s + 1,...,p. Let i=]

14 222 RAPHAEL BRU, LEIBA RODMAN, AND HANS SCHNEIDER and in B replace the Jrdan chain {Aigs:j =O,...,ks -I} by the Jrdan chain {AJu:j=O,...,k!(-I}. The result is a Jrdan basis fr M fr which the number f eigenvectrs f depth q -1 is t - P + 1. But, since B has t - P such eigenvectrs, this cntradicts ur assumptin n B. Hence Bl has the DP. Nw assume inductively that 1 < h ~ k and that Bh -1 has the DP. T prve that Bh has the DP, we cnsider t k.-1 O=1=W= E E aijajgp i=l j=k; (3.5) where k: = max{o, k j - h}, i = 1,..., t. We must prve that dpth(w) = min{dpth(a}gj):a ij *0, j = k;,... kj -1, i = I""Jt}. (3.6) If aij = 0 whenever j = k j - h, then w is a linear cmbinatin f elements f B h - l, and (3.6) fllws frm ur inductive assumptin. S assume that a 6j =1= 0 fr sme j = ks - hand 1 ~ s ~ t. Nte that it fllws frm ur assumptin n multiplicities that i=i,...,s (since the abve vectrs are eigenvectrs). and since h ~ 2 (and hence this vectr is nt an eigenvectr). Thus t prve (3.6) it is enugh t prve dpth( w) = min{dpth(algj): ajl * 0, j = k:..., k j - 2, i = I''''J t}. (3.7)

15 EXTENSIONS OF JORDAN BASES 223 T prve (3.7) we nte that t k;-2 Aw = E L aijaj+lgj i=lj=k; (and thus Aw =1= 0). Since AJ+lgj E B h - 1, j = k;~... k j - inductive assumptin yields 2, i = 1..., t, ur dpth(aw) = min{dpth{ai+lgj :a ii =1= 0, j = k[...,k j -2, i = I,...,t). (3.8) By Prpsitin 3.2, every chain has CP Hence, dpth(aw) = dpth( w) + 1, dpth(aj+lgj ) = dpth{a)gj) + 1, j=k;,...,k j -2, i,...,t. Hence (3.7) nw fllws frm (3.8), and thus Bh has the DP. By inductin, we btain that Bk has the DP, and since Bk = B, the result fllws. Prf f Therem 1.1. We may assume that A is nilptent. If the difference between the biggest and the smallest multiplicity f A is at least 2, then by Prpsitin 3.1 nt every A-invariant subspace is marked. Cnversely, assume that the multiplicities f A are equal t q and q -1, fr sme q ~ 2. In view f Prpsitins 3.2 and 3.3, every A-invariant subspace M has a Jrdan basis with the DP that als has the CP The therem nw fllws frm Therem 2.1. We can augment Therem 1.1 by the fllwing statement. THEOREM 3.4. Assume A is a nilptent. Then every A-invariant subspace is marked if and nly if there is q such that the index f every vectr in en is either q r q - 1. Therem 1.1 was cntained in an unpublished manuscript by the authrs dated July A related result (in the framewrk f slutins f Riccati equatins) was btained independently in [3].

16 224 RAPHAEL BRU, LEIBA RODMAN, AND HANS SCHNEIDER 4. PROOF OF THEOREM 1.2 We will actually prve a mre infrmative result. THEOREM 1.2'. The fllwing are equivalent fr a nilptent matrix A: (1) All multiplicities are equal (t q). (2) Fr all x E en" CO}, ht(x)+dpth(x) = q. (3) All invariant suhspaces are strngly marked. Prf. 0) ~ (2): Let C be a Jrdan basis in en. Then every element f C f height h has depth q - h, and (2) fllws because C has the OP (see the prf f the "if' part f Therem 2.1). (2) => (3): Clearly, (2) implies that every chain has the,cp Let W be an invariant subspace fr A. Since all Jrdan bases fr W cntain the same number f eigenvectrs, and by (2), an eigenvectrs have the same depth q - 1, it fllws that every Jrdan basis fr W satisfies the hyptheses f Prpsitin 3.3. Hence every Jrdan basis fr W has the OP. We nw btain (3) by Therem 2.1. (3) ~ (1); Suppse (1) is false, and x and y generate Jrdan chains f lengths q and r respectively, where r < q. Let u = Ar-1y - Aq-Ix and v = Aq-IX Then dpth(u) = dpth(v) = q -1, but dpth(u + v) = r -1. Hence the Jrdan basis {u, v} fr the invariant subspace span {u, v} des nt have the OP. By Therem 2.1, span{u, v} is nt strngly marked. We nw give a characterizatin f cnditin (1) in Therem 1.2' in terms f the Weyr characteristic. Recall that the Weyr characteristic f a matrix X crrespnding t eigenvalue A is the vectr (WI' w 2,..., Wd), ~here and d is the largest multiplicity f X crrespnding t A. j = 1,2,... d, PROPOSITION 4.2. The fllwing statements are equivalent fr a nilptent matrix A (we dente by d the largest multiplicity f A): (i) K(Ad-1)CR(A); (ii) the Weyr characteristic f A is (WI>'" > Wd). where W l = W 2 =." =Wd; (iii) all multiplicities f A equal d.

17 EXTENSIONS OF JORDAN BASES 225 Here R(A) dentes the range f A. Prpsitin 4.2 can be easily prved by inspecting the Jrdan fnn f A. We thank D. Hershkwitz fr reading the manuscript and suggesting several imprvements in the expsitin. REFERENCES 1 R. Bru and M. Lpez-Pellicer, Extensins f algebraic Jrdan basis, Glas. Mat. 20(40): (1985). 2 I. Ghberg, P. Lancaster and L. Rdman, Invariant subspaces f matrices with applicatins, Wiley, A. Pastr and V. Hernandez, The class f Hennitian and nnnegative definite slutins f the algebraic. Riccati equatin (preprint), presented at the Cnference n Riccati Equatins in Systems, Signals and Cntrl (S. Bittanti, A. J. Laub, J. C. Willems, eds.), Cm, Italy, Received 16 Nvember 1989; final manuscript accepted 29 May 1990