Æ !" #$ %&'()S( ) *+,-./ 01"2345 (67Æ89 :;< =>?@AB(CDE,FGHI7JKL MNOPQ 7RSTU(VWS( )XYZ-([)\ "] a 1 ;a 2 ;;a n+1 ^_`an+1CNbcdefg^hic#jkR lc"] x=[x 1 ; ;x n+1 ] t ^C n+1 mnop7h x j gNqrs"]KtuC n+1 #v9xwx op#j nyz{7dA^diag(a 1 ;;a n+1 )2Ka |}"~a j g2mna7A^S( )\ "A ^ S( ) XY"S( ) *+gXYZA 7RS h*+DZU!Vy S( )7S( ) *+vg "^5(` N7v; ¡z{"a¢` ¡7£¤g25(¥ ¦§ v¨©ª «¬"¦§W220036® ¯ °±<² a³´ t"µ¶5(` 7=>?9£v ·¡"5¸¹^(Cº»¼ ½7¾¿ÀÁf<9ÂÃÄÅ ¢CÆÇ"ÈÉ ¡vÊ" Ë*+=>?Ì ÍÎ:2ÆÏÅÐ 7£¤'()S( ) *+  ¡,(CÑÒ* ÓÔ"5ÕÖ^ÆÐ !C,×2Ø <® ¦ a t ÑÒ*ÐÙ (C-9-./ ÚÆ"ÆÐ ÛÜÝÞ vT(1)5C½ ßÎ(àÄq XY) (C-ám âã7 (2)Mirmanä å7æàç(CèrvéyS( ) êëìÖ íî ïð79(3)a¬ ¢Co ñXYTòó|}Zwô|}ò véyS( )Zy"
Hwa-Long Gau
and Pei Yuan Wu
Abstract
Inthissurveyarticle,wegiveanexpositoryaccountoftherecentdevelopmentson
thePonceletpropertyfornumericalrangesoftheoperatorsS( ). Itcanbeconsidered
as an updated and more advanced edition of the recent expository article published
intheAmericanMathematicalMonthlyby the secondauthoronthis topic. The new
informationincludes: (1)asimpli edapproachtothemainresults(generalizationsof
Poncelet, Brianchon{Ceva and Lucas{Siebeck theorems) in this area, (2) the recent
discovery of Mirman refuting a previous conjecture on the coincidence of Poncelet
curves and boundaries of the numerical ranges of nite-dimensional S( ), and (3)
some partial generalizations by the present authors of the above-mentioned results
fromthe unitary-dilationcontexttothe normal-dilationone and alsofromthe
nite-dimensional S( )to the in nite-dimensional.
||||||||||||||||||||||||
2000 Mathematics Subject Classi cation: Primary 15A60, 47A12; Secondary 14H99.
Key words and phrases: Numerical range, Poncelet property, unitarydilation.
ThisworkwaspartiallysupportedbytheNationalScienceCounciloftheRepublicof
Chinaunder research projects NSC-91-2115-M-008-011and NSC-91-2115-M-009-009
Inrecentyears,theresearchonthenumericalrangesof nitematricesand
bound-ed operators has been very active, thanks tothe biennialconvening of the WONRA
(Workshop on Numerical Ranges and Numerical Radii). (For more information on
this, check the webpage http://www.resnet.wm.edu/cklixx/wonra02.html.) One
area of investigations concerns the numerical ranges of the nite-dimensional
com-pressions oftheshift. Itwasdiscovered thattheboundariesoftheirnumericalranges
possessthePonceletproperty,meaningthatthereexistin nitelymanypolygonswith
thepropertythateachhas allitsverticesonthe unitcircleand allitssides tangentto
theasserted boundary. Thisyieldsanunexpectedlinkbetweenthe twentieth-century
subject of numerical range and some nineteen-century gems of projective geometry.
Anexpository accountof this developmentwasgiven in[35], whichexplainsthe
per-tinent results in a historical context. The purpose of this survey is to update this
previous account by providinga simpli edapproach and expounding the recent
dis-coveries. ChiefamongthelatteristheonebyMirmanthatnoteveryalgebraicconvex
curveintheopen unitdisc whichhas thePonceletpropertyarisesasthe boundaryof
the numericalrange of the asserted operator, thus refuting aprevious conjecture on
identifying such numerical ranges by the Poncelet property. We will also elaborate
on our recent attempts in generalizing the main results in this area to more general
contextssuchasgeneralconvexpolygonsinsteadofpolygonswithverticesontheunit
circleandgeneralcompressionsoftheshiftinsteadofmerethe nite-dimensionalones.
In Section2 below, we start with abrief review of the de nition and basic
prop-ertiesof numericalranges ofoperatorsonaHilbert space. Wealsodiscuss thenotion
ofdilationand itsconnectionwith numericalranges. Section3 thentreatsnumerical
rangesof nitematrices. Here theextra toolofKippenhahn curve provesveryuseful.
Itinvolvesthe point-linedualityoftheprojectiveplane. Section4considersthe
presented,eachofwhichhasitsmeritinexposingcertainpropertiesoftheirnumerical
ranges. Section 5 gives the main results on the Poncelet property for the numerical
ranges of the compressions of the shift on nite-dimensional spaces. There are three
of them: generalizations of the Poncelet porism (on the existence of in nitely many
interscribing polygons between twoellipses),Brianchon-Cevatheorem (on the
condi-tion for the tangent points of an inscribing ellipse of a triangle), and Lucas-Siebeck
theorem (on the relationbetween zeros of apolynomial and its derivative). We then
moveon tothe partial generalizationsof these results inSection 6.
2. Numerical Range
Let A be a (bounded linear) operator on a complex Hilbert space H. The
nu-merical range of A is the set W(A) fhAx;xi : x 2 H;kxk = 1g in the complex
plane,whereh;idenotes theinnerproductinH. Inotherwords,W(A)istheimage
of the unit sphere fx 2 H : kxk = 1g of H under the (bounded) quadratic form
x 7! hAx;xi. Some properties of the numerical range follow easily from the
de ni-tion. Forone thing, the numericalrange isunchanged underthe unitary equivalence
of operators: W(A) = W(U
AU) for any unitary U. It also behaves nicely under
the operation of taking the adjointof an operator: W(A
) =fz :z 2W(A)g. More
generally,this is even the case when taking the aÆnetransformation: if
f(x+iy)=(a 1 x+b 1 y+c 1 )+i(a 2 x+b 2 y+c 2 )
is an aÆne transformation of the complex plane C, where x;y and a
j ;b j and c j ;j =
1;2,are all real and the latter satisfy a
1 b 2 6=a 2 b 1
, and if we de ne f(A)to be
(a 1 ReA+b 1 Im A+c 1 I)+i(a 2 ReA+b 2 ImA+c 2 I); where Re A = (A+A )=2 and Im A = (A A
)=(2i) are the real and imaginary
of numericalranges, the reduction through aÆne transformations is a handy toolin
many situations.
The most importantpropertyof the numericalrangeis thatW(A) isalways
con-vex. This is the celebrated Toeplitz{Hausdor Theorem from1918-19 [31, 15]. Over
the years, therearenumerous proofsandgeneralizationsofthisfact. Theusualproof
is to rst reduce it to the case of 2-by-2 matrices (since the de nition of convexity
involves only two points at a time) and show that the numericalrange of the latter
is a closed elliptic disc or one of its degenerate forms (circular disc, line segment or
a single point). Indeed, if A =
a b
0 c
, then W(A) is the elliptic disc with foci a
and c and minor axis of length jbj. An easy proof of this is to reduce A to
0 1
0 0
viasome aÆnetransformationand check directlythatthe latterhas numericalrange
fz 2C :jzj1=2g (cf. [19]).
The numericalrange is abounded set, but isnot closed ingeneral. Forexample,
if S isthe (simple) unilateral shifton l 2 : S(x 0 ;x 1 ;)=(0;x 0 ;x 1 ;);
then W(S)equals the open unit discD =fz 2C :jzj<1g. However, ifthe operator
A acts on a nite-dimensionalspace, then W(A)is obviously closed and hence
com-pact. For an arbitrary operator A, the closure of its numerical range W(A) always
contains the spectrum (A). Hence the numerical range gives a rough estimate of
the locationof thespectrum. Thisisone of thereasonstostudy the numericalrange
and provides its main applications. If A is normal, then W(A) equals (A) ^
, the
convex hull of (A). Thus, in particular, if A is a normal ( nite) matrix, then
ty boundedconvexset isthenumericalrangeofsomeoperatoronaseparableHilbert
space. (Note that if nonseparable Hilbert spaces are allowed, then every such set is
the numerical range of some normal operator; compare [26].) Even more intricate
is to determine, for each positive integer n, the numerical ranges of operators on
an n-dimensional space. Although many necessary/suÆcient conditions are known,
a complete characterization is beyond reach at this moment. One condition on the
boundaryofthenumericalrangeisworthnoting. If4isaclosedconvexsubsetofthe
plane, then every nondi erentiable point of the boundary @4 of 4 has two distinct
supporting lines of 4 with angle less then such that the closed section formed by
them contains 4. Such a point is calleda corner of 4. Accordingto this de nition,
theendpointsofalinesegmentarecorners. AresultofDonoghue[8,Theorem1]says
that a corner of W(A) which also belongs to W(A) is a reducing eigenvalue of A.
The latter meansthat there isa nonzero vectorx suchthat Ax=x and A
x=x.
The proof of this makes use of the geometric fact that an elliptic disc which is
con-taining and contained inW(A) must bereduced toa linesegment. It follows that
ifA isann-dimensionaloperator,thenW(A)can have atmostn corners. This gives
acertainconstraintonthe shapeofthe numericalrange ofa nite-dimensional
oper-ator. Usingthe condition forthe equality case of the Cauchy{Schwarz inequality,we
may prove the analogous result that any point in W(A) satisfying jj = kAk is a
reducing eigenvalueof A.
Associated with the numerical range W(A) is the quantity w(A), the numerical
radius ofA,de ned by supfjzj:z 2W(A)g. Forexample,ifS isthe unilateralshift,
then w(S)=1, and if A is normal,then w(A)=supfjzj:z2 (A)g.
We say that the operatorA onspace H dilates toB on K or B compressesto A
if there is an isometry V from H to K such that A = V
form
A
. The notion of dilation and compression is closely related to that of
numericalrange. For one thing, the numericalrange itself can be described in terms
of dilation. Namely, for any operatorA, the numericalrange ofA is the same as the
set of complex numbers forwhich the 1-by-1matrix [] dilatesto A. On the other
hand,ifAisanoperatorwhichdilatestoB,thenW(A)iscontainedinW(B). Hence
ajudiciouschoiceofanicelybehavedB canyieldusefulinformationonthenumerical
range of A. One type of dilation which will be fully exploited in our derivations in
Sections 5 and 6 is the unitary dilation of contractions. The classical result in this
respect is Halmos's dilation: every contraction A (kAk 1) can be dilated to the
unitary operator 2 4 A (I AA ) 1=2 (I A A) 1=2 A 3 5
(cf. [14, Problem 222 (a)). With more care, the unitary dilation can be achieved
in a most economical way: if A is a contraction on H, then A can be dilated to
a unitary operator U from H K
1 to H K 2 with K 1 and K 2 of dimension-s d A dimran (I AA ) 1=2 and d A dimran (I A A) 1=2 , respectively, and,
moreover, in this case d
A
and d
A
are thesmallest dimensionsof such spaces K
1 and K 2 . Here d A and d A
are calledthe defectindices ofthe contractionA. They provide
a measure on how far A deviates from the unitary operators and play a prominent
role in the unitary dilationtheory.
Propertiesofnumericalrangesofoperatorsarediscussedin[14,Chapter22];those
for nitematricesare in[16,Chapter 1]. Thetwoclassicmonographs[4]and [5]treat
thenumericalrangesofelementsofnormedalgebras;themorerecent[13]emphasizes
applications tonumerical analysis.
can be exploited to yield special toolswhich are not available for general operators.
OnesuchtoolisthecharacteristicpolynomialofthepencilxRe A+yImAassociated
with any matrix A. This can be utilized in two di erent ways to yield W(A) or its
boundary. One is via Kippenhahn's result that the numerical range of A coincides
withtheconvexhulloftherealpartofthedualcurveofdet(xRe A+yIm A+zI)=0.
In this way, the classical algebraiccurvetheory can bebroughtto bear onthe study
here. On the other hand, a parametric representation of the boundary @W(A) can
alsobeobtained fromthelargesteigenvalue ofcosRe A+sinIm Ayielding useful
informationon W(A). Here we givea brief accountof both approaches.
Let CP 2
be the complex projective plane consisting of all equivalence classes
[x;y;z] of ordered triples of complex numbers x;y and z which are not all zero.
Two such triples [x;y;z] and [x 0 ;y 0 ;z 0 ] are equivalent if x =x 0 ;y=y 0 and z =z 0
for some nonzero . The point [x;y;z] (z 6= 0) in homogeneous coordinates can be
identi ed with (x=z;y=z) in nonhomogeneous coordinates. On the other hand, the
point (u;v) becomes [u;v;1] inhomogeneous coordinates. In this way, C 2
is
embed-ded in CP 2
. If p(x;y;z) is ahomogeneous polynomialof degree d inx;y and z, then
the set of points [x;y;z] in CP 2
satisfying the equation p(x;y;z)=0 is analgebraic
curve of order d. If C issuch a curve, then its dualC is de ned by C =f[u;v;w]2CP 2 :ux+vy+wz =0 is atangent lineof Cg: In this case, C
is also an algebraic curve of order at most d(d 1) and d is called
the class of C
. It is known that the dual of C
is C itself. The point[x
0 ;y 0 ;z 0 ] isa
focus of C if it isnot equal to [1;i;0]and the lines through [x
0 ;y 0 ;z 0 ] and [1;i;0]
are tangent to C at points other than [1;i;0]. In general, if a curve is of class d
and isde ned by anequationwith real coeÆcients,then ithas d real fociand d 2
d
p
A
(x;y;z)=det(xRe A+yImA+zI
n )
and let C(A) denote the dual curve of p
A
(x;y;z) = 0. Since p
A
is a real
homoge-neouspolynomialofdegreen, the curveC(A)isgiven by arealpolynomialofdegree
at most n(n 1), is of class n, and has n real foci [a
j ;b
j
;1];j = 1;;n, which
correspond exactlyto the n eigenvalues a
j +ib
j
of A. The connection of C(A) with
the numerical range W(A) is provided by a result of Kippenhahn [18]: W(A) is the
convex hull of the real points of the curve C(A), namely, the convex hull of the set
fa+ib 2C : a;b 2R;ax+by+z =0 istangent to p
A
(x;y;z)=0g. Kippenhahn's
result can be easily veri ed by noting that x = max (Re (e i
A)) is a supporting
line of W(Re(e i
A)) for any real . Since it can be shown that @W(A) contains
only nitely many line segments, the above result implies that @W(A) is piecewise
algebraic, that is, it isthe union of nitely many algebraiccurves.
There is another way to make the above to be more revealing. For any
nonemp-ty compact convex subset 4 of the plane, there is a natural parametrization of its
boundary @4. For any ;0 2, let L
be the ray from the origin which has
inclination fromthe positive x-axis,and letM
be the supporting line of 4 which
isperpendicular toL
. If d() isthe signed distance fromthe origin toM
, then @4
can be \parametrized" by ()=(x();y()), where
x() = d()cos d 0 ()sin; y() = d()sin+d 0 ()cos:
Itcanbeshownthatd()isdi erentiableforalmostallandisequaltomaxfRe (e i
z):
z 24g. In particular, if 4=W(A) forsome operator A, then
d() = max (Re (e i
A))
A
inz has d()asa zero. Asan example,if A isthe 3-by-3 matrixdiag (1;i;0),then
d()= 8 > > > < > > > : cos if 0 4 or 3 2 2; sin if 4 ; 0 if 3 2 ;
and the naturalparametrization of @W(A) (=the triangularregion with vertices 1;i
and 0) isgiven by ()= 8 > > > < > > > : 1 if 0< < 4 or 3 2 < <2; i if 4 <<; 0 if < < 3 2 :
In particular, this shows that the natural parametrization is not a parametrization
in the usual sense: it does not traverse the line segments on the boundary; but the
convex hullof its imageequals @4.
4. Compression of the Shift
Compressions ofthe shift are aclass of operators studiedintensively inthe 1960s
and '70s. Playing arole analogous to the companion matrix in the rationalform for
nite matrices, they are the building blocks in the \Jordan form" (under
quasisim-ilarity) for the class of C
0
contractions. The whole theory is subsumed under the
dilation theory for contractions onHilbert spaces developed by Sz.-Nagy and Foias.
Thestandard referenceisthemonograph[30]; amore completeaccountofthe theory
of C
0
contractions isgiven in [3].
We start by noting that the unilateral shift S has another representation as
(Sf)(z) = zf(z) for f in H 2
, the Hardy space of square-summable analytic
roinvariantsubspaces of S are ofthe form H 2
forsome innerfunction ( is inner
if it is bounded and analytic on D with j (e i
)j = 1 for almost all real ). The
compression of the shiftS( ) is the operator onH( )H 2
H 2
de ned by
S( )f =P(zf(z));
where P denotes the (orthogonal) projection from H 2
onto H( ). Thus S( ) is the
operator in the lower-right corner of the 2-by-2 operator matrix representation of S
as 2 4 0 S( ) 3 5 on H 2 = H 2 H( ):
Thisclass ofoperatorswas rststudiedby Sarason[29] andhas been underintensive
investigationoverthe past 35years. In particular,it isknown that kS( )k=1;S( )
is cyclic (there is a vector f(= 1 (0) ) in H( ) such that W
fS( ) n
f : n 0g=
H( )), and its commutant fS( )g 0 (fX on H( ) : XS( ) =S( )Xg) and double commutant fS( )g 00 (fY on H( ): YX = XY for every X in fS( )g 0 g) are both equal toff(S( )):f 2H 1
g. The inner function is the minimal function of S( )
inasensesimilar tothe minimalpolynomialofa nite matrix,that is,itissuch that
(a) (S( ))=0,and(b) isafactor ofanyfunctionf inH 1
forwhichf(S( ))=0.
AnoperatorAis(unitarilyequivalentto)acompressionofthe shiftif andonlyif itis
acontraction,bothA n
andA n
convergeto0inthestrongoperatortopology,andthe
defect indicesd
A and d
A
are bothequaltoone. Itfollowsfromtheseconditions that
the compression of the shiftis irreducible, that is, itcan have no nontrivial reducing
subspace.
For nitematrices,thecharacterizationofcompressionsoftheshiftineven easier:
Aissuchanoperatorif andonlyifitisacontraction,ithas noeigenvalueofmodulus
one and d
A
(z)= n Y j=1 z a j 1 a j z ; wherea j
'sare theeigenvalues ofAinD. We letS
n
denotethe classof such matrices.
An example inS
n is J
n
, the n-by-nnilpotentJordan block
2 6 6 6 6 6 6 6 6 6 4 0 1 1 0 3 7 7 7 7 7 7 7 7 7 5 ;
withthecorrespondinginnerfunction (z)=z n
. BytheresultsinSection2,matrices
in S
n
admit unitary dilations on an (n+1)-dimensionalspace. For this reason, S
n
-matricesare called matricesadmittingunitary borderingorUB-matrices by Mirman
(cf. [21, 22, 23]). Since S( ) is de ned by its minimal function , we infer that for
any npointsa
1
; ;a
n
inD (notnecessarilydistinct)there isamatrixinS
n
, unique
uptounitaryequivalence,withthea
j
'sasitseigenvalues. Amorespeci cdescription
of a matrix inS
n
with eigenvalues the a
j 's is given by [a ij ] n i;j=1 , where (1) a ij = 8 > > > < > > > : a j if i=j; [ Q j 1 k=i+1 ( a k )](1 ja i j 2 ) 1=2 (1 ja j j 2 ) 1=2 if i<j; 0 if i>j:
This matricialrepresentation was rst discovered by Young [36, p. 235] (cf. also[28,
p. 201], [21, Theorem 4] and [11, Corollary 1.3]). In particular, it follows that S
2
consists of 2-by-2matrices which are unitarily equivalentto amatrix of the form
2 4 a (1 jaj 2 ) 1=2 (1 jbj 2 ) 1=2 0 b 3 5
with a and b in D. There is another representation for matrices in S
n
which is
useful for our discussions in Section 5. If A is in S
n
diag (1; ;1;a) with 0 a < 1. The equality WAW
= (WU)D shows that A is
unitarily equivalent to (WU)D, a matrix of the form
(2) [f 1 f n ] whose columns f j satisfy kf j k = 1 for 1 j n 1;kf n k < 1 and f j ? f k for
1j 6=k n. Conversely, a matrix of the above form with noeigenvalue of
modu-lusone isin S
n .
>From the information we have so far on the compressions of the shift, we can
already deduce certain properties of their numerical ranges. Let A be a matrix in
S
n
. Then W(A) must be contained in the open unit disc D. This is because if in
W(A)is suchthat jj=1(=kAk),then itwillbea reducing eigenvalue of A, which
contradicts the irreducibility of A. On the other hand, by Donoghue's result and
the irreducibilityof A,wemay deduce that the boundary ofW(A) isadi erentiable
curve. In the subsequent sections, we willdiscuss other ner propertiesof W(A).
5. Poncelet Property
The recentestablishmentofalinkbetweenthe numericalrangesofmatricesinS
n
and some classical geometric results from the 19th century was achieved by Mirman
[21, 23,22, 24] and the present authors [9, 10, 11,12]. Here we give a brief account
of this development.
Our rst result has to do with a geometric theorem of Poncelet. In this treatise
[27] of 1822, there is contained the following result,called Poncelet's porism or
Pon-celet's closure theorem: is C and D are ellipses in the plane with C inside D, and if
there is one n-goncircumscribed about and inscribed in D, that is, the n-gon has n
the assertion says that some property (the existence of a circumscribing-inscribing
n-gon)eitherfailsor,ifitholdsforone instance,succeeds in nitelymanytimes. Itis
a closuretheorem since, fromany point on D,we draw atangentlineto C, which
intersects D atanother point,then repeatthis process bydrawing tangentlinesfrom
successivepointsobtained inthis fashion,andobtaintheresultingclosedn-gonwhen
thenthtangentlinereachesbackto. Vieweddynamically,thisgivesacon guration
ofrotatingn-gonswith di erentshapesbut allsharingthis circumscribing-inscribing
property. Sincethe appearance ofthis result,ahuge literaturehas been developedto
its explanation, exposition and generalization. A comprehensive survey of this topic
can be found in [6]. We may normalize the outer ellipse D asthe unit circle @D via
some aÆne transformationand the inner ellipse C istransformed into one inD with
then-Ponceletproperty. More precisely,forn3,wesaythatacurve inD has the
n-Poncelet propertyif for every point on@D there isann-gon which circumscribes
about ,inscribesin@D andhas asavertex. Itisnaturaltoask whetherthere are
curves other than ellipses in D which also have the n-Poncelet property. The next
theorem provides more examples.
Theorem5.1. Forany matrixAinS
n
andpoint on@D, thereisaunique
(n+1)-gonwhich circumscribesabout @W(A),inscribes in @D andhas as avertex. Infact,
such(n+1)-gonsP arein one-to-onecorrespondencewith(unitary-equivalence
class-es of)unitary dilationsU of Aonan (n+1)-dimensionalspace,underwhichthen+1
vertices of P are exactly the eigenvalues of the corresponding U.
This theoremappearedin[21, Theorem1]and[9,Theorem2.1]. The easypartof
theproofistoshowthatevery(n+1)-dimensionalunitarydilationofamatrixAinS
n
has distincteigenvalueswhichforman(n+1)-goninscribed in@D andcircumscribed
details, we resort tothe matrix representations (1)and (2) forA togive the speci c
(n+1)-dimensionalunitary dilationsU. If A isrepresented as in(1), then U can be
taken as[b ij ] n+1 i;j=1 , where (3) b ij = 8 > > > > > > < > > > > > > : a ij if 1i;j n; [ Q j 1 k=1 ( a k )](1 ja j j 2 ) 1=2 if i=n+1 and 1j n; [ Q n k=i+1 ( a k )](1 ja i j 2 ) 1=2 if j =n+1 and 1in; Q n k=1 ( a k ) if i=j =n+1;
for some in @D. Here acts as a parameter for the unitary dilations U. On the
other hand, if A isas (2), then U can be
(4) 2 4 f 1 f n 1 f n g 0 0 a kf n k 3 5 ; where jj=1;a=(1 kf n k 2 ) 1=2 >0 and g = 8 < : (a=kf n k)f n if f n 6=0;
any unit vector orthogonal to f
1 ; ;f n 1 if f n =0:
Both (3)and (4) can be used to prove that the (n+1)-gonswith vertices the
eigen-values of U coverall pointsof @D (the latteris in[9, Theorem 2.1]).
Theorem 5.1 yields additional properties for the numerical ranges of matrices in
S
n .
Corollary 5.2. Let A be a matrix in S
n
. Then
(a) W(A) is contained in no m-gon inscribed in @D for mn,
(b) w(A)>cos(=n),
Here (a) is an easy consequence of the (n+1)-Poncelet property of @W(A), (b)
followsfrom(a),(c)isaconsequenceof,besidesthePonceletproperty,theinterlacing
of theeigenvalues ofRe A andRe U for (n+1)-dimensionalunitarydilationU of A,
and nally(d) follows from(c) by way of Kippenhahn's result. All assertions except
(d) are in [9].
If A is in S
n
, so is e i
A for any real . Hence the eigenvalues of Re (e i
A) are
all distinct by Corollary 5.2 (c). The curves
j ;j = 1;;n, described by j () = (x j ();y j ()) with x j () = j ()cos 0 j ()sin; y j () = j ()sin+ 0 j ()cos; where j
() isthe jthlargest eigenvalue of Re(e i
