The Electronic Journal of Linear Algebra.
A publication of the International Linear Algebra Society.
Volume 6, pp. 20-30, March 2000.
ISSN 1081-3810. http://math.technion.ac.il/iic/ela
ELA
ON THERELATION BETWEEN THENUMERICAL RANGE ANDTHE JOINTNUMERICAL RANGE
OF MATRIX POLYNOMIALS P. J. PSARRAKOS
y
AND M. J. TSATSOMEROS z
Abstrat. It isshown thatthe numerialrange,NR[P()℄, ofa matrix polynomialP() = Am
m
+:::+A 1
+A 0
onsistsoftherootsofallsalarpolynomialswhoseoeÆientsorrespond totheelementsoftheonvexhullofthejointnumerialrangeofthe(m+1)-tuple(A
0 ;A
1 ;:::;A
m ). Moreover, the elements of the joint numerialrange that giverise to salarpolynomials witha ommonrootbelongingtoNR[P()℄formaonnetedset. Thelatterfatisusedtoexaminethe multipliityofrootsbelongingtotheintersetionoftherootzonesofNR[P()℄. Alsoan approxima-tionshemeforNR [P()℄isproposed,intermsofnumerialrangesofdiagonalmatrixpolynomials.
Keywords.jointnumerialrange,matrixpolynomial,rootzone AMSsubjet lassiations. 15A60,47A12
1. Introdution. The numerial range of a matrix polynomial and the joint numerialrangeofaset ofmatries(usually takento betheoeÆientsofamatrix polynomial)arebothnotionsthatgeneralizethelassialnumerialrangeofamatrix. The former is a one-dimensional generalization and the latter a multi-dimensional generalization. Wewillontinuepreviouseortsfoundinourreferenestostudythe relationsbetweenthesesetsanddisovertheirproperties. Theonludingparagraph ofthissetioninludesabriefdesriptionofourresults,requiringthefollowingbasi terminologyandnotation.
LetM n
(C)denotethealgebraof nnmatriesovertheomplexeldandlet P()=A
m
m
+:::+A 1
+A 0 (1)
beamatrixpolynomial,where A j
2M n
(C)(j=0;1;:::;m),A m
6=0. Thepositive integers m and n are referred to as the degree and the order of P(), respetively. Thesalar
0
2C is alled aneigenvalue ofP() iftheequation P( 0
)x=0hasa nonzerosolutionx=x
0 2C
n
. ThusthespetrumofP()istheset [P()℄=f2C : detP()=0g:
As detP() isapolynomialofdegreeat mostnm, [P()℄ iseither equaltoC or itonsistsofatmostnmomplexnumbers. Thenumerial rangeofP()istheset
NR[P()℄=f2C : x
P()x=0 forsomenonzerox2C n
g;
Reeivedbytheeditorson6May1999. Aeptedforpubliationon27February2000.Handling editor: BryanL.Shader
y
DepartmentofMathematisandStatistis,UniversityofRegina,Regina,Saskathewan,Canada S4S0A2(panosmath.uregina.a).
z
ELA
JointNumerialRange 21 whihislosedandontains[P()℄. NotiethatthenotionofNR[P()℄generalizes thelassialnumerialrangeof amatrixA,thatis,theonvexset
NR[I A℄=fx
Ax : x2C n
;x
x=1g: Also notie that
0
2 NR[P()℄ if and only if 0 2 NR[I P( 0
)℄. It is known that NR[P()℄ is not neessarily onneted and that it is bounded if and only if 062NR[I A
m
℄; seeLiandRodman[6℄.
Thejointnumerial rangeofP()isthesetofomplex(m+1)-tuples JNR [P()℄=f(x
A
0 x;x
A
1
x;:::;x
A m
x) : x2C n
; x
x=1g:
Thejoint numerialrange, beingaontinuous image ofthe unit sphere,is ompat and onnetedbutnotneessarily onvex;see Binding andLi[1℄. Itsonvexhullis denoted byCofJNR[P()℄g. Observethat
NR[P()℄=f2C : m
m
+:::+ 1
+ 0
=0; ( 0
; 1
;:::; m
)2JNR[P()℄g: Consider now the following two subsets of C: the set of roots of all m-degree salar polynomials whose oeÆients orrespond to elementsof the joint numerial rangeofm+1givenmatries,andthesetofrootsofallm-degreesalarpolynomials whose oeÆients orrespond to elements of the onvexhull of the joint numerial rangeof the samem+1matries. As wewill see, boththese sets oinide and are equal to the numerial range of the assoiated matrixpolynomial, despite the fat thatthejointnumerialrangeisnotneessarilyonvex(Proposition2.1). Moreover, the elements of the joint numerial range assoiated with salar polynomials that havea ommon root belonging to the numerialrange, form aset that is not only nonemptybutalsoonneted(Proposition2.2). Whetherornotthemultipliitiesof rootsof salarpolynomialswith oeÆientsfrom thetwosetsabovealso arryover isanopen(andseeminglysubstantial)problemwithimpliationsinthefatorization ofmatrixpolynomials. Somepartialresultsinthisregardanbefoundinsetion3. Finally, in setion4 wedesribeaproess to approximate the numerialrange ofa matrixpolynomialusingthe(joint)numerialrangesofdiagonalmatries.
2. The onvex hull of the joint numerialrange. Notiethat ifQ()isa matrixpolynomialofdegreem, then
JNR[P()℄JNR[Q()℄ =) NR[P()℄NR[Q()℄:
Moregenerally,givenasetGofomplex(m+1)-tuples suhthat JNR[P()℄G, NR[P()℄ f2C :b
m
m
+:::+b 1
+b 0
=0; (b 0
;b 1
;:::;b m
)2Gg: (2)
Thefollowingstrengtheningof(2)holdswhenG=CofJNR[P()℄g.
Proposition2.1. For everymatrixpolynomialP()of degree mwehave NR[P()℄=f2C :
m
ELA
22 P.J.PsarrakosandM.J.Tsatsomeros Proof. Let
)2CofJNR[P()℄g. ByCaratheodory'sTheoremapplied to C
Sinethereexist unitvetorsx j
2C n
suhthat q wehavethat
andthus(3)beomes
Sinealltheomplexnumbersx Theoppositeinlusionfollowsfromthedisussionpreedingthisproposition.
We ontinue with an interesting geometri feature of the relation between NR[P()℄ and JNR[P()℄. Suppose that exista
0 Wemaythendenethesetofomplex(m+1)-tuples
T ByProposition2.1,ifT
1 (
0
ELA
JointNumerialRange 23 Proposition 2.2. Let P() = A
Proof. ThesetT 1
. Itisshownin LyubihandMarkus[8℄thattheset L
isonnetedforn3. Consequently,ifweonsidertheontinuousfuntion :fx2C indeedaonnetedset.
In the speial ase when P() is self-adjoint, that is, the oeÆient matries are Hermitian, JNR[P()℄ onsists of real (m+1)-tuples. Therefore for every! 2 NR[P()℄\R weandenetherealhyperplane SimilarlytoProposition2.2,itanbeshownthatifP()isannnmatrixpolynomial withn3,thenT
1 (!)
R
\JNR[P()℄isonnetedforevery!2NR[P()℄\R. 3. Root zones of the numerial range. Let P() be a matrix polynomial as in (1), whose numerial range is bounded, that is, 0 62 NR[I A
m ℄. We an always number the roots of x
P()x = 0 to obtain a sequene of funtionals
(x) on theunit sphereS, havingranges 1 tively. Wewillreferto
1
as`rootbranhes'and usesthem in identifying onditionsthat are equivalentto theroots ofx
P()x=0 beingsimpleateverypointofS.
If there exists a nonzero x 2 C n
suh that x
P()x = 0 has multiple roots, thensomerootzoneshavenonemptyintersetionandthusNR[P()℄ hasfewerthan m onneted omponents. In what follows, we will onsider the onverse, namely, whether thenonemptyintersetion ofsomeoftheroot zonesneessarilyimpliesthe existeneofmultiple roots; seeMaroulasandPsarrakos[9,Theorem3.1℄forthe self-adjointase. For that purpose, wewill use the notionof the jointnumerial range andtheresultsoftheprevioussetion. Wewillalsoneedtorestritourattentionto aompatsubsetofS,wherethefuntionals
1 (x);
2
(x);:::; m
(x)arewelldened andontinuous. Weexpandonthisrequirementnext.
Certainly, the roots of x
P()x = 0 depend ontinuously on the oeÆients x
A
j
x,andthusonx. ThisfatanbededuedfromOstrowski[10,pp.334{335℄by onsideringtheompanionmatrixofthepolynomial
m
+b m 1
ELA
24 P.J.PsarrakosandM.J.Tsatsomeros where
b j
= x
A
j x x
A
m x
(j=0;1;:::;m 1): Inspiteoftheontinuousdependeneoftherootsofx
P()x=0onx,itmaynotbe possibletodenetheaforementionedfuntionals
1 (x);
2
(x);:::; m
(x)ontinuously on thewhole unit sphere S. This situation is illustrated by the following example. Considerthematrixpolynomial
P()=I 2
0 2 0 2
andletx=(ose i
;sin) T
parameterizeS. Wethenhavethat x
P()x= 2
sin(2)e i
: Thus, if we requireontinuityof the roots
1 (x);
2
(x)as funtions of x, then it is easytoverifythattheyannotbedenedglobally onS.
When the roots of x
P()x = 0 are simple for every x 2 S, a unique global denition of eah
j
(x) as a ontinuous funtional on S is possible and is indeed obtainedin Lyubih[7℄. Otherwise,to proeedwithourgoal,weneedtomodifythe notionofaroot branh
j
(x),using in essenetheargumentsin [7,pp.55{56℄. We will dene ontinuousfuntionals
j =
j
(x)on aertainompat subsetM
of S, using notionsfromdierentialgeometryfor whih ourgeneralreferene isHelgason [4℄. WeonstrutM
under theassumption thatthepolynomial L(;x)x
P()x2C[;x 1
;x 2
;:::;x n
℄
has no multiple irreduible fators. Let D(x) be the disriminant of L(;x) with respettotheindeterminate,thatis,D(x)istheresultantofthepolynomialL(;x) and itsderivative
L(;x). If D(x)0in thepolynomialringC[;x 1
;x 2
:::;x n
℄, then L(;x) has a repeated fator not in C[x℄; f. Walker [11, p. 25℄). Therefore, underourassumption,D(x)isanon-zeroelementofC[x℄.
Considernowtheset U =[C
n
nf0g℄nfx2C n
: L(;x)=
L(;x)=0 forsome2Cg =fx2C
n
nf0g:D(x)6=0g:
The set U is open and is the omplement of an algebrai hypersurfae in C n
. A omplexalgebraihypersurfaehasreal odimension2in thewhole spae. Thus by [4,p. 346,Corollary12.5℄,U isarwiseonneted. It islearthat U isdensein C
n . Thereforetheset
U 1
=fx2C n
:x
x=1;D(x)6=0g
is an arwiseonneted dense open subset of theunit sphereS. The set U 1
ELA
JointNumerialRange 25 andeneonM
0
analytifuntionals 1
; 2
;:::; m
beingrootsofx
P()x=0,via amethodsimilartothatmentionedin[7℄. Denoteby
0
theoveringmapofM 0
onto U
1
. Foreveryp2M 0
, thesefuntionalssatisfy i
(p)6= j
(p)for1i<jm. For everypairp;q of pointsof M
0
satisfying 0
(p)= 0
(q), thereis apermutation of f1;2;:::;mgforwhih
i
(q)= (i)
(p) (i=1;2;:::;m): DenenowanequivalenerelationonM
0
asfollows. pq ifandonlyif
0 (p)=
0
(q) and i
(p)= i
(q) (i=1;2;:::;m): Dene now M as the osetspae M
0
= . Then M is also a overingspae of U 1
. Let denote the overing map of M onto U
1
. The funtionals j
are well dened onM. Foreverypairp;q of M for whih (p)=(q) andp6=q, thereis an index 1 i m with
i
(p) 6= i
(q). For everyx 2 U 1
, the number L of points p 2 M for whih (p) = x is the same and not greater than m!. Thus, M is an analyti manifoldwiththenaturalRiemannianstruture. Denotebydthemetrifuntionon M orS. Asearlier,usingtheresultsin[10,pp.334{335℄,weaninfat deduethat thefuntionals
j
areuniformlyontinuousonM. Takinganarbitrary>0,asS is ompat,therearepointsy
1 ;y
2 ;:::;y
`
inS suhthat S=
` [ j=1
fy2S : d(y;y j
)<=2g:
SineU 1
isdenseinS,thereexistx 1
;:::;x `
2U 1
withd(x j
;y j
)<=2(j=1;2;:::;`). Thus
U 1
= ` [ j=1
fx2U 1
: d(x;x j
)<g:
For eah j, there exist points p j;1
;p j;2
;:::;p j;L
of M for whih (p j;k
) = x j
(k = 1;2;:::;L)andp
j;s 6=p
j;t
for1s<tL. Then,
M= ` [ j=1
L [ k =1
fx2M : d(x;p j;k
)<g:
Sine >0is arbitrary,the spaeM istotally bounded. Theompletion M
of M isompat. Byuniformontinuity,thefuntionals
j
aredened andontinuouson M
.
The map an be extended to a ontinuous map of M
onto the unit sphere S. Based on the above disussion and notation, we will in the sequel refer to the root zonesof NR[P()℄ (whereP() isamatrixpolynomialasin (1)with bounded numerialrange)asthesets
ELA
26 P.J.PsarrakosandM.J.Tsatsomeros Eahrootzone
j
isaompatsubsetoftheGaussianplaneC. For
0
2NR[P()℄onsidernowaompatsetF M
thatontainsavetorx 0 denethesets
V whereT
1
Theorem 3.1. Consider the nn matrix polynomial P() = A m
be distint root zones of NR[P()℄. If
thereexistx 1 ing the proof of Proposition 2.2, we an show that T
F
(!) is a onneted subset of JNR[P()℄. Moreover,
T
where these three sets are ompatsubsets of JNR[P()℄. Consequently, V i;F
), then there always exist p;q2M
(q)=!. However,fortheexisteneofthevetorx 0
as intheabovetheorem, pandq mustbeonnetedbyaontinuousurvein
fx2S:x
P(!)x=0g\M
; afat impliedbytheremainingassumptionsofTheorem3.1.
Theorem3.2. ConsiderthennmatrixpolynomialP()=A m
ELA
JointNumerialRange 27 Proof. AsintheproofofTheorem3.1,thesets
V i;F
(!) and [ k 6=i
V k ;F
(!)
arenonemptyandompatsubsetsofF M
. Moreover,theset
T F
(!)=V i;F
(!)[ 2 4
[ k 6=i
V k ;F
(!) 3 5
isaompatandonnetedsubsetofJNR[P()℄. Hene,
V i;F
(!)\ 2 4
[ k 6=i
V k ;F
(!) 3 5
6=;;
thatis,there existsx 0
2F suhthat i
(x 0
)= k
(x 0
)=! forat leastonek6=i. 4. Polyhedra andthe jointnumerialrange. Considertwomatrix polyno-mials P() = A
m
m
+:::+A 1
+A 0
and Q() = B m
m
+:::+B 1
+B 0
, not neessarilyof the sameorder. IfJNR[P()℄ JNR[Q()℄ orifCofJNR[P()℄g CofJNR[Q()℄g, thenbyourdisussionin setion2,NR[P()℄NR[Q()℄. Letus further assume NR[P()℄ and NR[Q()℄ are bounded. Then forany(
0 ;
1 :::;
m ) in JNR[P()℄, theequation
m
m
+:::+ 1
+ 0
=0has at leastoneroot in ev-eryonneted omponentof NR[P()℄. A similar statement istrue for JNR[Q()℄ and theonneted omponents of NR[Q()℄. Thus NR[Q()℄ has at mostasmany onnetedomponentsasNR[P()℄.
Inviewof theabove,onsider nowtwodiagonalmatrixpolynomialsD 1
() and D
2
()suhthat
JNR[D 1
()℄JNR[P()℄JNR[D 2
()℄: (5)
Let#NR[P()℄denotethenumberofonnetedomponentsofNR[P()℄. Fromthe preedingdisussionitfollowsthat
#NR [D 1
()℄#NR [P()℄#NR[D 2
()℄: (6)
Observethat theordersofD 1
()andD 2
()areirrelevantaslongas(5)issatised. Moreover, (6) gives an approximation of the number of onneted omponents of NR[P()℄ that an be quite useful, espeially when the order n of P() is a lot greaterthanitsdegreem.
Wedoknowthatthejointnumerialrangeofadiagonalmatrixpolynomialisa onvexpolyhedron;see[1℄and[9℄. Conversely,ifKC
m+1
isaonvexpolyhedron withnverties
ELA
28 P.J.PsarrakosandM.J.Tsatsomeros thenK=JNR[D()℄, where
D()=diag( 1;m
;:::; n;m
) m
+:::+diag( 1;1
;:::; n;1
)+diag( 1;0
;:::; n;0
): Lemma 4.1. LetT C
m
be a nonempty ompat set andW asubspae of C m suh that W \K 6=; for every onvex polyhedron K C
m
that ontains T. Then CofTg\W 6=;.
Proof. LetK bethefamilyofall onvexpolyhedrathat ontainT. Notiethat K isintersetionlosed. SineT isompat,wehavethat
CofTg= \ K2K
K :
Dene F = fW \K : K 2 K g. By assumption, all members of F are ompat, nonemptyandonvex. Moreover,ifW \K
j
2F (j=1;2;:::;p),wehave p
\ j=1
(W \K j
)=W\( p \ j=1
K j
)2F
sine T
p j=1
K j
2 K . Hene by Helly's Theorem (see Danzer, Grunbaum, and Klee [3℄),
T K2K
(W\K)6=;. It followsthat CofTg\W =( T
K2K
K)\W 6=;: Thefollowingresultisageneralizationof[9,Theorem1.1℄.
Theorem4.2. LetP()=A m
m
+:::+A 1
+A 0
beannnmatrixpolynomial. Then
[ NR[D
1
()℄=NR[P()℄= \
NR[D 2
()℄; (7)
where the union [intersetion℄ is taken over all diagonal matrix polynomials D 1
() [D
2
()℄ofdegreemfor whih JNR[D 1
()℄JNR[P()℄JNR [D 2
()℄. Proof. ForeverydiagonalmatrixpolynomialD
1
() with JNR[D
1
()℄JNR[P()℄itislearthatNR[D 1
()℄NR[P()℄. Thus [
JNR[D1()℄JNR[P()℄ NR[D
1
()℄NR[P()℄:
Inaddition,forevery=( 0
; 1
;:::; m
)2JNR[P()℄weantakeD 1
()= m
I m
+ :::+
1 I+
0
I sothatJNR[D 1
()℄=fg. Hene [
JNR[D 1
()℄JNR[P()℄ NR[D
1
()℄NR[P()℄
and the rst equation in (7) holds. Forthe seond equation, notie that for every diagonalpolynomialmatrixD
2
()ofdegreemforwhihJNR[D 2
()℄JNR[P()℄, wehaveNR[D
2
()℄NR[P()℄. Thus NR[P()℄
\
JNR[D2()℄JNR[P()℄ NR[D
ELA
JointNumerialRange 29 Letnow
0
2
\ JNR[D
2
()℄JNR[P()℄ NR[D
2 ()℄:
Fromthedisussioninsetion2(see(4))itfollowsthatT 1
( 0
)\JNR[D 2
()℄6=;for everydiagonalmatrixpolynomialD
2
()satisfyingJNR[D 2
()℄JNR[P()℄. Hene fromLemma 4.1,
T 1
( 0
)\CofJNR[P()℄g6=;; andfromProposition2.1,T
1 (
0
)\JNR [P()℄6=;. Sothereexistsunitx 0
2C n
suh that
(x 0 A
m x
0 )
m 0
+:::+(x 0 A
1 x
0 )
0 +x
0 A
0 x
0 =0; whihinturnimplies
NR[P()℄
\
JNR[D2()℄JNR[P()℄ NR[D
2 ()℄;
ompletingtheproof.
Observation 4.3. In(7)weanreplaeJNR[P()℄byitsonvexhull.
Observation 4.4. Theorem 4.2givesamethod of approximationofNR[P()℄ by numerialranges of diagonal matrixpolynomials. With the help of Proposition 2.1, weandevelopa seond method as follows. Considertwosequenes of onvex polyhedrafK
(i) t
g(i=1;2andt=m+2;m+3;:::)sothat K (i) t
hast verties, K
(1) t
CofJNR[P()℄gK (2) t
(t=m+2;m+3;:::); andlastlysothatbothK
(i) t
onvergetoCofJNR[P()℄gast!1. IfweletD (i) t
() for i=1;2bett diagonalmatrix polynomials assoiatedwith K
(i) t
, respetively, thenforalltm+2,
NR[D (1) t
℄NR[P()℄NR[D (2) t
℄ andbothNR[D
(i) t
℄ onvergeto NR[P()℄ ast!1. Inotherwords,NR[P()℄an be approximated by numerial ranges of diagonal matrix polynomials of the same degreeandappropriatelylargeorder.
TheresultsinBinding, Farenik,and Li[2℄suggestanothermethodof assoiat-ing numerialranges of matrixpolynomialsto numerial ranges ofdiagonal matrix polynomialsviadilations.
Aknowledgments. Weaknowledgewiththanksananonymousrefereefor gen-erouslyprovidinguswiththeonstrutionoftheset M
ELA
30 P.J.PsarrakosandM.J.Tsatsomeros REFERENCES
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