On Stability of Invariant Subspaces of Matrices Author(s): A. C. M. Ran and L. Rodman Source: The American Mathematical Monthly, Vol. 97, No. 9 (Nov., 1990), pp. 809-823 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2324747 . Accessed: 12/12/2014 16:57 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp
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ofInvariantSubspacesofMatrices On Stability TheNetherlands Amsterdam, A. C. M. RAN,VnijeUniversiteit, andMary L. RODMAN, TheCollegeof William hisM.Sc.andhisPh.D. at theVrijeUniversiteit C. M. RAN received wereM. A. KaashoekandI. In bothcaseshisthesisadvisors inAmsterdam. his Ph.D. in 1984he spenta yearin the US, Gohberg.Afterreceiving mainlyat VirginiaTech, on a grantfromthe Niels StensenStichting at the VrijeUniversiteit Since 1985he has been working (Amsterdam). and control are operatortheory, matrices, again.His researchinterests theory.
ANDRE
University Tel-Aviv from received hisPh.D. inmathematics positionsat Tel-AvivUniversity in 1978.Since thenhe held permanent and The Collegeof Williamand Mary (Israel),ArizonaStateUniversity of Calgary(Canada),Vrije and visiting positionsat University (Virginia), (San Diego). His of California and University Universiteit (Amsterdam) matrices, (operators, are mainlyin linearmathematics researchinterests linearsystems andcontrol). LEIBA RODMAN
1. Introduction.In thispaper we studythe behaviorof invariantsubspaces of matrices,under small additiveperturbationsof these matrices.In particular,we willbe interestedin studyingthe rate of convergence(in termsof the normof the perturbation)of the invariantsubspace of the perturbedmatrix.It turnsout that fromthatof eigenvectorsonly.In some thisrate of convergencecan be different cases the rate of convergenceof invariantsubspaces is actuallymuch betterthan that of eigenvectorsif one has the choice of an invariantsubspace for the Also, new phenomenaappear; forexample,the rate of converperturbedmAtrix. ifall nearbyinvariantsubspaces of the perturbedmatrixare gence can be different considered.In thispaper we develop the conceptsof stabilitythatreflectthe rate of convergenceof invariantsubspaces. We point out some new phenomena that occur in this situationand prove partial results concerningcharacterizationof invariantsubspaces with a given rate of convergence.Many questions here are open and we indicatesome of them. It is well known that the eigenvalues of an n X n (complex or real) matrix depend continuouslyon the matrix.The behaviorof eigenvectorshas been extensivelystudiedfromthe pointofviewof applicationsin numericalanalysis[W,GvL]. In fact,the behaviorof eigenvectorscan be basicallydescribedas follows:it fallsin eitherone of the followingthreeclasses: 1) hopeless: consider,for example, the eigenvector[1] of the matrix[? 0. Perturbingthe matrixto [?
0] withE a small numberwe see thatthe eigenvector 809
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810
ANDRE
C. M. RAN AND LEIBA RODMAN
[November
of theperturbed matrix is farawayfrom[I . Of course,thereis nothing special
o]
about [ot; anyothereigenvector of [0 is justas hopeless. 2) continuousbut bad: the standardexamplehere is the n X n matrix . 0 1 0 0
.
...
0
...
0 Jn(6)~~ 80
...
..
1
..
0 O
j we have the followingeigenvectorsof J (8): y1(8) = (1,?j, 2 n,..., -1)T, 1,.. ., n where ej is givenby Ej = I8I1"nexp(2wij/n).Clearly,forthe vectorsyj we have llyj- ell = 0(1Ill/n) where e1 = (1,0, ... , o)T is the eigenvectorof Jn.(Here and everywherein the paper we use the Euclidean normforthe columnvectors.) We call it bad because the rate of convergenceof the eigenvectorsof Jn(8) to e1, is 0(E111/n),whichis slow forlarge n. 3) continuousand good: the case of an n x n matrixwith n differentand well-separatedeigenvalues.In thiscase, as we shall also see later,the perturbation of the eigenvectorsis of the same order of magnitudeas the perturbationof the matrix. It is natural to classifyalong these lines the behavior of general invariant subspaces. (Clearly, an eigenvectorcan be identifiedwith the one-dimensional subspace it generates).Given an n x n complexmatrixA, a subspace MC Cn is called A-invariantif AM c M. Here c stands for (not necessarilyproper) inclusion. Every invariantsubspace is spanned by a set of eigenvectorsand generalizedeigenvectorsof A (recall that X = 0 is a generalizedeigenvectorof A belongingto the eigenvalueof A if(A - AI)'X = 0 forsome j > 1). In the next sectionwe shall recall some of the resultson stabilityof invariant subspaces,beforegoingto the main topicof thispaper,whichis treatedin sections threeand four.
2. Stable invariantsubspaces. The stabilitypropertiesof invariantsubspaces underperturbations of the matrixhave been studiedextensively in the last decade [BGK, CD, KvdMR,GLR2, GR]. A source where all thismaterialis gathered(and which includes much more) is [GLR2]. We start with the definitionof stable invariantsubspaces. We shall denote the set of invariantsubspaces of a matrixA by Inv(A). Let A be an n x n complexmatrixand let M E Inv(A). M is called stable if for all E > 0 there is a 8 > 0 such that lB - All < 8 implies the existence of N E Inv(B) withII M- PNII < E. Here PM (resp., PN) is the orthogonalprojection
This content downloaded from 130.37.164.140 on Fri, 12 Dec 2014 16:57:10 PM All use subject to JSTOR Terms and Conditions
1990]
ON STABILITY
OF INVARIANT
SUBSPACES
OF MATRICES
811
on M (resp., N), and IXII denotes the operator norm of a matrix X, i.e., withllXlI beingtheusualEuclideannorm,JI(Xl,.. ., Xn)T112 lXII= max1I11,.=IIXXIl, The main result on is the stability following theorem. En.J1Lj2. -
THEOREM 2.1. [BGK, CD]. Let A be an n x n complexmatrixand M E Inv(A). ThenM is stableifand onlyifforeach eigenvalueA ofA withdimKer(A - AI) > 1 eitherM n Ker(A - AI)n is (0) or it is Ker(A - AI)n. For eigenvaluesA ofA with dimKer(A - AI) = 1 the space M nl Ker(A - AI)n is an arbitrary A-invariant subspacecontainedin Ker(A - AI)n.
We denote byKer X the kernelof the matrixX. ObservethatKer(A - AI)n is just the root subspace correspondingto A, i.e., the subspace spanned by all eigenvectorsand generalizedeigenvectorsbelongingto the eigenvalueA. Equivalently,Ker(A - AI)n is the largestA-invariantsubspace N withthe propertythat the restrictionA/N has A as its onlyeigenvalue. This stabilityresultjust tells us when M is continuouslydependenton A. In case M is stable both the "continuousand good" and the "continuousbut bad" cases mentionedin the introduction can occur. To distinguish betweenthe twowe need another definition.An invariantsubspace M E Inv(A) is called Lipschitz stable if thereexistpositivenumbersK and 8 such that B - All < 8 impliesthe existenceof N E Inv(B) with IIlM- PNlI S K IIB - All. Obviously,this defines preciselywhat we mean by "continuous and good." The followingtheorem describessuch subspaces. THEOREM 2.2 [KvdMR]. Let A be an n X n complexmatrix,let M E Inv(A). Then M is Lipschitzstable if and only iffor each eigenvalueA of A eitherM n Ker(AI -A)A = (0) orM D Ker(AI -A)n.
As the proof is illustrativefor our approach to anotherproblem(in the next section)we shall give an outlineof the proofhere. In additionto [GLR2], [GK] is an excellentsource forthe backgroundtheoryused. Proof. Firstsuppose that M satisfiesthe conditionof the theorem.Then M is a spectralsubspace; i.e., thereis a contourF in C such that A - AI is invertiblefor A E F and
M = Im[ 2
(AI-A)
1dA,
wherewe denote by Im X the range(i.e., the columnspace) of the matrixX. The matrix(1/27ri)fr(AI - A)-' dA is a (not necessarilyorthogonal)projection.Now take 8 small enough such that AI - B is invertibleon F for lB - All < 8. Then put N= Im[27 if(AI
-B)YdA]d
The matrix(1/2wri)fr(AI- B)-' dA is again a projection.At thispointwe need the fact (the proof of which can be found,e.g., in [GLR]) that given subspaces M, N c C? withorthogonalprojectionPM and PN on M and N, respectively, we have
IIPM- PNII< IIQM - QNII
(2.1)
where QM and QN are arbitrary (not necessarilyorthogonal)projectionson M and
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812
ANDRE
[November
C. M. RAN AND LEIBA RODMAN
N. Using thisfactwe obtain IPm-
PNI
0. For thiswe need the followinglemma. arbitrarily
3.3. Let p and q be complexnumbersdepending on complex polynomially Zn) suchthatIp(z)I< C1IIzII,Iq(z)l < C2IIZII2 for I[zilsmall enough,wherethepositiveconstantsC1 and C2 are independentof z. Given any 8 > 0 all solutionsw of the cubic equation W3 + pw + q =0 will satisfyIwI< forllzllsufficiently small(and C3 > 0 independent C311ZII2/3-5, ofz). LEMMA
z = (z, ...., parameters
Proof. Recall that w is foundas follows.Put 3
31
U= 1/-
+
(q2
V=
()3,
+-
(2) -
(2
()3
Then w = u + v, wherethecubicrootsin u and v are chosensuchthatuv = -p73 (see e.g., [M] pp. 671-676 or [BS] pp. 160-164). We now consider two cases separately. - forsome 8 > 0. (1) IP(Z)I> C41jIZI4/3 (2) Ip(z)l< C511ZII413-8forsome 8 > 0. (We denote by Cj positiveconstantsindependentof z.) In case (1), we write 3q
(q)
-
)
+
Then W=
3
13 3
3
-(3
+(3-q=
3
+
3
1q
1 q2
- 9 A1 33
32
5 q3 1623 3
)
q 3
small IIzil.Indeed, Clearlythisseries converges,as jql C 6 11Z114
C6
IIZI1-35-=> o* as IIzil
0.
smallenough.Now for lizil lizil-* 0, and consequently,lql < 1/31
This content downloaded from 130.37.164.140 on Fri, 12 Dec 2014 16:57:10 PM All use subject to JSTOR Terms and Conditions
1990]
OF INVARIANT
ON STABILITY
SUBSPACES
817
OF MATRICES
small llzIlwe have forsufficiently
IqI
IWI< c
123
(3.6)
On the otherhand, (q\
q
2 )+
2 -
q
(P3
+
q(1
+
2
3)
small So for l zll sufficiently
i
= 1X31
4 p3 2
+~~~~~~~~
27qP +
)
)
=
1 p3
27 q
= C8Z11I2-385
?811
So 1132/31 >
Cq2IZIl4/32S,
Thisfinishes case (1). and thus(using(3.6)), IWI< C10 IIZII2/3+28. In case (2) we estimate 13
q
U13 < l-
q 2
+
-
+
p3C1IlI7128 A C C 1 12-lzll'3/25 so
-
lIUlA C1311Z12/3-1/2.
Likewise,also lvI < C14
IIZI12/3-1/2.
Thus, lwj
0 + (a2/4) + b, we have jai < C1IIA -A't11/3-, forsome arbitrarily small.Now, pick A1and A2twoeigenvaluesof A' providedIIA - A'II is sufficiently such that JA1+ A21= Let x = (1, X1, X2, X3)T and y = then
()
|
=t
t2
a
2
-
|
C A -A'll
(1, y1, Y2, y3)T
f
-a A
-b(
be the correspondingeigenvectors;
|
i )1
and a similar formulaholds for (1, y1,y2,y3)T replacing A1 by A2. Assuming
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818
A1= A2, consider (x
-
ANDRE
C. M. RAN
y)/(A,
-
AND
LEIBA
A2) = u. We have:
0
to0
1 + Al A2 1 1 A + A1A2+A2
x -y = AI A
[November
RODMAN
-at -a
(A,
1 1+A
Aj
_b|
K)
So, if we put M' = span{x, y} = spanfx, u} we have M' = Im[4, R=
[
LA1Ia
l-b - A,(Al + A2) + Ala bAl- c Al(Al + AlA2 + A2 - aAl - aA2 -b)
aAA2 I-
where Al + A2
A1+ 12 + A2 -aA
)a
)
A2) - b
Now let
Q R
O]
-[
04
Then Q'2 = Q' is a projectiononto M', while Q is a projectiononto M. Clearlyfor small IIQ' - Qll = IIRII< CIIA - A'll1/3-, as the dominant IIA - A'll sufficiently termin R is in its upper rightcorner.Then (using(2.1)) we obtain
IIPM- PMiI < IIQ- Q'Il < CIIA -A'll We have provedthatforany positivee thereexistsa constantC > 0 such that every4 x 4 matrixA' withdistincteigenvalueshas an invariantsubspace M' such that IIM - PMIIl< CIIA - AII1/3-. This does not mean yetthat M is 3(1 - 3-Y '-stable,because we have to account formatricesA' withmultipleeigenvalues.This can be done as follows: Since the set of matriceswith distincteigenvaluesis dense, given any 4 x 4 matrixA' thereis a sequence of 4 x 4 matrices{AI 1}= withdistincteigenvalues such that limm-OOA'= A'. By the already proved statementfor every m there existsan Am-invariant subspace Mm,such that -,. IIPM- PM',,11 < CIIA -Al 11I/3
(3.7)
Now we make use of the fact that the set of all subspaces in C' is a complete compact metricspace in the metricgiven by IIPM- PNII, where M, N are subspaces in C". Moreover, a subspace N which is a limit (in this metric) of a = 1 can be describedas sequence of subspaces {Nm}m N
=
{X
E
Cl thereis a sequence ofvectors{Xm}l='
such thatXmE Nmforall m and lim Xm= m-
A proofof thisfactcan be found,forexample,in [GLR1] or [GLR2]. Returningto the sequence of subspaces {Mm,}m=introducedabove, choose a convergingsubse1 witha limitM'. Using the descriptionof a limitgivenby (3.8) we quence {Mm,_}k= easilyverifythat M' is A'-invariant.Passing to the limitin (3.7) (with m replaced by Mk) we obtain the desiredinequality
IIPM-
PM'Il < CIIA
We have provedthat M is 3(1
-
-
AqlI1/3E.
301-stable foreveryE > 0.
This content downloaded from 130.37.164.140 on Fri, 12 Dec 2014 16:57:10 PM All use subject to JSTOR Terms and Conditions
1990]
819
ON STABILITYOF INVARIANTSUBSPACESOF MATRICES
Clearly,fore small enough3(1 - 3e)-l < 4. This showsthatalthoughM is not stronglya-stable forany a < 4, it is a-stable foreverya with3 < a < 4. 4. The stabilityindex. It follows from Theorem 3.1 that every stable Ainvariantsubspace is stronglya-stable (and hence also a-stable) with a equal to the heightof A. The heightof A is definedas the size of the largestJordanblock in the Jordancanonicalformcorresponding to eigenvaluesA of A withdimKer(A - AI) = 1. (If for all eigenvalues A of A one has dimKer(A - Al) > 1 we put the heightof A equal to one.) We nextdefinethe stability indexof a stable A-invariantsubspace M to be the infimumof all a > 0 such that M is a-stable, and this numberwe denote by s(M, A). To avoid trivialities we will applythisnotiononlyto the non-trivial (i.e., different from(0) and the whole space) stable invariantsubspaces. In thissectionwe studythe stabilityindex of a stable invariantsubspace. Here too, many questions remain unsolved. We do give some bounds and estimates, whichallow us to computethe stabilityindexin special cases. A standardreduction(see e.g., Section 15.3 in [GLR] or Section 8.1 in [BGK]) establishesthe followingfact. PROPOSITION4.1. Let A1,... , Ar be thedistinct eigenvaluesof then X n matrix A such that dimKer(A - Ail) = 1. Put Ri = Ker(AiI - A)y. Then for any stable M E Inv(A) we have s(M, A) = maxi< i < rs(M n Ri, A/Ri), whereA/Ri stands fortherestriction ofA to itsinvariantsubspaceRi.
The nextpropositiongivesupper and lowerbounds fors(M, A). PROPOSITION4.2. s(M, A) is less thanor equal to theheightofA, and s(M, A) > 1, provided(0) 0 M =#Cn.
Proof The factthat s(M, A) does not exceed the heightof A is a consequence of the previouspropositionand the remarksin the firstparagraphof thissection. To show s(M, A) > 1, assume the contrary.By Proposition4.1 we can assume withoutloss of generalitythat A = Jn is the n x n Jordan block with zero eigenvalue and M = span{el, ..., ek}, k < n. Let Jn(e) be the matrixobtained from Jn by puttinge > 0 in the left lower corner. By assumption,there is a = subspace M(?) such that IIPM- PME.)II< CIIA - A(?)11/a C?-/ Jn(c)-invariant for e small enough. Let x(?) = (1, e.. ., ci1) E M(?) be an eigenvectorof Jn(e) correspondingto its eigenvalue ei which is an nth root of e. Then, since
x(?)
E
M(?), we have (PM
PM(8))X()
|| =
IIPMX(e)
-
x(_)II ? C?l/alIX(E)II.
Thus,in particularconsideringthe (k + 1)stcoordinateof PMX(c) -
we have
|?|k-n
=
lJilk
n/k, and thatactuallyone can say s(k, n) > n/iO where jo = min{jIpj(E,. . ., #k)=# 0 for all choices of different 1,... .,k out of El, . . ., n)}.However,these bounds are not verysatisfying. The nexttheoremis of more interestand is our main result.To state it we need the followingdefinition.Given two positiveintegersk, n with k < n, we say that the pair (k, n) is admissibleif no sum of k distinctnth roots of unityis equal to zero. In particular(1, n) is admissibleforall n > 2. 4.4. If n is prime,then(k, n) is admissibleforevery1 < k < n.
PROPOSITION
* + 1 is Proof Use the fact that the cyclotomicpolynomialXn - + x n-2 + irreducibleover the fieldof rationalnumbers(see, e.g., [McD]). Assume now that the primitiveroot 4 = cos(2T/n) + i sin(2r/n) satisfies an equation of type .
MI
'M2
+
..
++
+4Mk
0
=
(n-1
> m2
> ml
>
...
> mk
Then the cyclotomic polynomial must be a divisor of Xml +
are integers).
>O
-
XM2 +
+Xmk
(see
e.g., the proof of Theorem 38.1 in [McD]), which is possible only if k = n. This provesthe proposition. p, 0 < p < k - 1, THEOREM 4.5. Let k < n/2. Assumethereis a smallestinteger such that (q, n - p) is admissiblefor all integersq with k - p < q < k. Then s(k, n) > n-p. Proof For e small and positive,let A(?) be the matrixobtained fromJnby puttinge in the (n - p, 1)-entryin place of zero. The eigenvalues of A(?) are zero and the (n - p)th roots of e, call them 1 . . n. Yi
p- The eigenvectors corresponding to
(1, 77i,
P71
0 10 .., ?)
,*,
71n-pare
* I Yn_p= (1 71-p 7n I ..**S n-p
0 ..0)
and thereis a Jordanchain correspondingto the eigenvalue0: e2
e-een-p+l
-
cen-p+2
...,
ep -een,
whereby eq we denote the unit coordinatevectorwith 1 in the qth positionand zeros elsewhere.Consequently,everyk-dimensionalA(e)-invariantsubspace has the form
N = span{yi,*..., yiq,el-
en-p+l
e2
-
en_P+
.2
ek-q
-
en-p+(k-q)}
forsome q(k - p < q < n - p; q < k) and some choice of the vectorsYil,*. , Yiq among Y1, *- , Yn-p. Introduce also the subspace M = span{yi,, ., Yiq, .
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1990]
ON STABILITY
el, e2, ... , ek/q}.
OF INVARIANT
SUBSPACES
821
OF MATRICES
Clearly,
IIPN- PMll < Kj_(4.1) forsome constantK1 independentof E. Aftersome elementarytransformations of the set of spanningvectorsforM, we obtain that M is spannedby el, e2, . . ,ekq and by the vectors ek-q+t ? kr1jte-q+2+
t = 1,..., q
1
wheres = n - p - (k - q). (It followsfromour assumptionson k and p that s is positive.)DenotingbyMO the k-dimensionalJ -invariantsubspace,it is easilyseen now that - PM| IIPMO
Pspan{f1 ,ff}
=
Pspan{ek
q
+ 1,
., ek}ii
Consider the n x q matrixW with columns fl,.. ., fq: W = [f1,f2,... , fql, and observe that the rows k - q + I throughk in W formthe Vandermondematrix
F1
1
'77i
.
WO,-
q1 ,qi1
11
. . .
71i
T
,
.
niq gIi
...
q-l
ni2
(4.2)
whichis invertible.A computation(see e.g., [GLR, Section 16.5]) shows that the
matrixWV1 has the(k + 1,q)-entry equal to qj, +
. * * +71i
Now use the assumptionthat(q, n - p) is an admissiblepair (since k - p < q < k). It followsthat thereis a constantK > 0 such that IA1 + * +Aql > K for any choice of q (n - p)th rootsof unityA1,. . . Aq Then also K,il +
-
* +
iq I
> K
(4.3)
* i?1l/(n-P)
for any choice of the q numbers .j . . , Ni . Now consider the last column in Call this vector X. Clearly XEE span{f1,... , fq} and X is of the form WWJJ1. wherethe firstk - 1 entriesare zero. Then (4.2) (O, . . ., O, L+ +7i,. )jq . . gives
|PMO PM>
lXII Pspan{f1_fq}X1
11lXIIX
ekil >
Pspan{ek_q, .,
1
11ii 1 il +
..
+
ek}A
ql.
For E small enoughwe obtain by using(4.3) that iiPM0 -~PMII
forsome constantK2
> 0
> K
I/(n -_
p)
independentof e. Taking into account (4.1), we get IIPN - Pmill
>
K31EI1I(n-p)
(4.4)
forany k-dimensionalN E Inv(A()), where K3 > 0 is independentof both e and N. Inequality(4.4) shows that MO E Inv(J,) cannot be a-stable for a < n - p whichproves s(n, k) > n - p, and our theoremis proved.
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822
ANDRE
C. M. RAN AND LEIBA RODMAN
[November
As a corollaryto Theorem 4.5 and Proposition4.4 we have 4.6. Let r be the smallestnumbersuch that n - r is prime. Then COROLLARY s(k, n) > n - rforall k(1 < k < n - 1). In particular,s(k, n) = n forprimen. It follows,for example, that s(2, 4) > 3, and combiningwith our resultsobtained in the example in the previoussectionwe see that s(2, 4) = 3. In terms of the stabilityindex we can say preciselywhen a-stabilityfor a subspace M impliesstronga-stability.To state the resultwe need the following notion.Given an n x n matrixA and M E Inv(A) we introducethe heightofA withrespectto M as follows.Let (A1,... . A.) be the set of all distincteigenvaluesof A withdimKer(Ai -A) = 1, and let a = {ill < i < r,O o M rnKer(AiI -A)n 0 Ker(AzI - A)y}. Then the heightof A withrespectto M, denoted by h(M, A) is the heightof AIN where N is the sum of the subspaces Ker(AI - A)n, i E oF. Again, the heightof A withrespectto M is definedto be 1 if o- is empty.With thisdefinitionwe have the followingresult. are equivalentfora stableA-invariant 4.7. Thefollowingtwostatements THEOREM subspaceM: (a) s(M, A) = h(M, A) a-stableifand onlyifM is a-stable. (b) For any a: M is strongly Proof The case when cr= 4 is easy. In thiscase M is the sum of rootsubspaces and hence Lipschitzstable and stronglyLipschitzstable by Theorem 2.2, hence Theorem 4.7 followsfromthe definitionof h(M, A). So we assume cf / 4 from now on. Clearly, both s(M, A) and h(M, A) as well as a-stabilityand strong a-stabilityforM depend onlyon M n N and on the restrictionAIN, whereN is as ?D J,, M = M1,E * M, where Mi is above. So we can assume A = J? with0 #Mi = Ker(AiI - A)n,. We also can arrangeit so that n1 > n2 Ji-invariant --* > nr, in whichcase h(M, A) = n,. > Further,M is a-stable or stronglya-stable withrespectto A ifand onlyif each Mi is a-stable or stronglya-stable withrespectto J,. Now suppose (a) holds.Then if M is a-stable a mustbe largerthanor equal to nl. Then certainlya > ni for 1 < i < r. By Theorem 3.1 it then followsthat M is stronglya-stable. As stronga-stabilityimplies a-stabilityalways,thisproves(b). Conversely,suppose (b) holds for any a. Then, if M is a-stable, it must be stronglya-stable and hence Theorem 3.1 implies that a > ni for each ni. Thus, a > h(M, A). This shows s(M, A) > h(M, A). But clearlyM is h(M, A)-(strongly) stable. So s(M, A) = h(M, A). Note thatwe use here a sharpeningof Proposition 4.2. The theoremis proved. We have introducedthe notion of stabilityindex and have produced some bounds on the stabilityindex.These bounds allowed us to findthe precisevalue of the stabilityindexin some cases. It is an open problemto findthe stabilityindexin the generalcase. 5. Open problems.We summarizehere some open problems(includingthose mentionedalreadyin the text)thatarise naturallyin the contextof a-stability. Problem5. For a given a, describe(in termsof the intersectionsof M withthe root subspaces Ker(A - AI)n) all a-stable subspaces M of a matrixA.
This content downloaded from 130.37.164.140 on Fri, 12 Dec 2014 16:57:10 PM All use subject to JSTOR Terms and Conditions
1990]
ON STABILITY
OF INVARIANT
SUBSPACES
OF MATRICES
823
Problem5.2. Describe the relationshipbetween a-stabilityand stronga-stability.In particular,a more concreteproblemis: When does a-stabilityimplystrong a-stability?Again, the answer should be given in termsof intersectionsof the invariantsubspace in questionwiththe root subspaces. Problem5.3. Find the stabilityindexof everystable A-invariantsubspace M, in terms of the dimensionsof Ker(A - AI)' and M rnKer(A - AI), for every eigenvalueA of A. In particular,is the stabilityindexalwaysa positiveinteger? The solutionof Problem5.3 hingesupon solutionto the following: Problem5.4. Find s(k, n) forall positiveintegersk and n, 1 < k < n - 1. Again,partialanswersand bounds fors(k, n) are foundin the text.It follows,in particular,that s(k, n) = n for all (k, n) with 1 < k < n - 1; 2 < n < 5; except that s(2, 4) = 3. Also, s(k, 6) = 6 if k = 1, 5 and 5 < s(k, 6) < 6 for k = 2, 3, 4.
Hence the firstconcreteopen questionis:
Problem5.5. Find the values of s(2, 6) = s(4, 6) and s(3, 6). We are gratefulto Prof.M. A. Kaashoek forhis interestin the materialof this Acknowledgments. paper and forhis suggestionto writeit up. REFERENCES Arnold, V. I.: On matrices depending on parameters,Uspehi Math. Nauk, 26 (1971) 101-114 [Russian]. of matrixand operator Bart,H.; Gohberg,I. and Kaashoek, M. A.: Minimalfactorizations [BGK] functions,OT1, BirkhauserVerlag,Basel, 1979. Bronshtein,I. N. and Semendyayev,K. A.: A guide-bookto mathematicsfortechnologists [BS] and engineers,MacMillan, New York, 1964. Campbell, S; and Daughtry,J.: The stable solutionsof quadratic matrixequations, Proc. [CD] AMS 74 (1979), 19-23. Gohberg, I. C. and Krein, M. G.: Introductionto the theoryof linear nonselfadjoint [GK] operators,Translationsof Mathematical Monographs,Vol. 24, American Mathematical Society,Providence,R.I., 1970; reprinted1988. Gohberg,I., Lancaster,P. and Rodman, L.: MatrixPolynomials,Academic Press, 1982. [GLR1] Gohberg,I., Lancaster,P. and Rodman, L.: InvariantSubspaces of MatriceswithApplica[GLR2] tions,J. Wiley& Sons, New York, etc., 1986. Gohberg, I. and Rodman, L.: On distance between lattices of invariantsubspaces of [GR] matrices,LinearAlgebraAppl., 76 (1986) 85-120. Golub, G. H. and Van Loan, C. F.: MatrixComputations,JohnsHopkins UniversityPress, [GvL] Baltimore,1983. [KvdMR] Kaashoek, M. A., van der Mee, C. V. M. and Rodman,L.: Analyticoperatorfunctionswith compact spectrumII: Spectral pairs and factorization,IntegralEquations and Operator Theory,5 (1982) 791-827. MacDuffee,C. C.: An Introductionto AbstractAlgebra,J. Wiley& Sons, New York, 1940. [McD] Meyers Handbuch uber die Mathematik,BibliographischesInstitut,Mannheim, Wien, [M] Zurich,1972. [German]. Ran, A. C. M. and Roozemond,L.: On stronga-stabilityof invariantsubspaces of matrices, [RR] collection,OT40, BirkhauserVerlag,Basel, 1989,427-435. in: The Gohberganniversary Wilkinson,J. H.: The AlgebraicEigenvalue Problem,ClarendonPress, Oxford,1965. [WI [A]
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