DTIC ADA548786: Bounds for Eigenvalues of ArrowheadMatrices and Their Applications to HubMatrices andWireless Communications

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BOUNDS FOR EIGENVALUES OF ARROWHEAD MATRICES AND
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Lixin Shen, and Bruce Suter
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14. ABSTRACT
This paper considers the lower and upper bounds of Eigenvalues of arrow-head matrices. We propose a parameterized
decomposition of an arrowhead matrix which is a sum of a diagonal matrix and a special kind of arrowhead matrix whose
Eigenvalues can be computed explicitly. The Eigenvalues of the arrowhead matrix are then estimated in terms of Eigenvalues of the
diagonal matrix and the special arrowhead matrix by using Weyl’s theorem. Improved bounds of the Eigenvalues are obtained by
choosing a decomposition of the arrowhead matrix which can provide best bounds. Some applications of these results to hub
matrices and wireless communications are discussed.
15. SUBJECT TERMS
Matrix Calculus, Signal Processing, MIMO System, Wireless Communications, Eigenvalue
16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF RESPONSIBLE PERSON
ABSTRACT OF PAGES Bruce Suter

a. REPORT b. ABSTRACT c. THIS PAGE 19b. TELEPHONE NUMBER (Include area code)
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EURASIPJournalonAdvancesinSignalProcessing
Volume2009,ArticleID379402,12pages
doi:10.1155/2009/379402
Research Article
Bounds for Eigenvalues of Arrowhead Matrices and Their
Applications to Hub Matrices and Wireless Communications
LixinShen1andBruceW.Suter2
1DepartmentofMathematics,SyracuseUniversity,Syracuse,NY13244,USA
2AirForceResearchLaboratory,RITC,Rome,NY13441-4505,USA
CorrespondenceshouldbeaddressedtoBruceW.Suter,[email protected]
Received29June2009;Accepted15September2009
RecommendedbyEnricoCapobianco
This paper considers the lower and upper bounds of eigenvalues of arrow-head matrices. We propose a parameterized
decomposition of an arrowhead matrix which is a sum of a diagonal matrix and a special kind of arrowhead matrix whose
eigenvaluescanbecomputedexplicitly.Theeigenvaluesofthearrowheadmatrixarethenestimatedintermsofeigenvaluesof
thediagonalmatrixandthespecialarrowheadmatrixbyusingWeyl’stheorem.Improvedboundsoftheeigenvaluesareobtained
bychoosingadecompositionofthearrowheadmatrixwhichcanprovidebestbounds.Someapplicationsoftheseresultstohub
matricesandwirelesscommunicationsarediscussed.
Copyright©2009L.ShenandB.W.Suter. This is an open access article distributed under the Creative Commons Attribution
License,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperly
cited.
1.Introduction recently in [5–8] such that the constructed arrowhead
matrix has the pregiven eigenvalues and other additional
In this paper we develop lower and upper bounds for requirements.
arrowhead matrices. A matrix Q ∈ Rm×m is called an Our motivation to study lower and upper bounds of
arrowheadmatrixifithasaformasfollows: arrowheadmatricesisfromKungandSuter’srecentworkon
⎡ ⎤
the hub matrix theory [9] and its applications to multiple-
D c
Q=⎣ ⎦, (1) inputandmultipleoutput(MIMO)wirelesscommunication
ct b systems.Amatrix,sayA,isahubmatrixwithmcolumnsifits
firstm−1columns(callednonhubcolumns)areorthogonal
where D ∈ R(m−1)×(m−1) is a diagonal matrix, c is a vector to each other with respect to the Euclidean inner product
in Rm−1, andb is a real number. Here the superscript and its last column (called hub column) has a Euclidean
“t” signifies the transpose. The arrowhead matrix Q is norm greater than any other columns. Subsequently, it was
obtained by bordering the diagonal matrix D by the vector shown that the Gram matrix of A, that is, Q = AtA, is
c and the real number b. Hence, sometimes the matrix an arrowhead matrix and its eigenvalues could be bounded
Q in (1) is also called a symmetric bordered diagonal by the norms of the columns of A. As pointed out in
matrix. In physics, arrowhead matrices have been used to [9–11], the eigenstructure of Q determines the properties
describe radiationless transitions in isolated molecules [1] of wireless communication systems. This motivates us to
andoscillatorsvibrationallycoupledwithaFermiliquid[2]. reexaminestheseboundsoftheeigenvaluesofQandmakes
Numerically efficient algorithms for computing eigenvalues them sharper. In [9], the hub matrix theory is also applied
and eigenvectors of arrowhead matrices were discussed in to the MIMO beamforming problem by comparing k of m
[3]. The properties of eigenvectors of arrowhead matrices transmittingantennaswiththelargestsignal-to-noiseratio,
werestudiedin[4],andasanapplicationoftheirresults,an including the special case where k = 1 which corresponds
alternative proof of Cauchy’s interlacing theorem was given toatransmittinghub.Therelativeperformanceofresulting
there.Theexistenceofarrowheadmatriceswasinvestigated systemcanbeexpressedastheratioofthelargesteigenvalue
1
2 EURASIPJournalonAdvancesinSignalProcessing
of the truncated Q matrix to the largest eigenvalue of the Lemma1. LetQ∈Rm×mbeanarrowheadmatrixoftheform
Q matrix. Again, it was previously shown that these ratios (1),whereD =diag(d1,d2,...,dm−1)∈R(m−1)×(m−1),b∈R,
couldbeboundedbytheratiosofnormsofcolumnsofthe andc=(c1,c2,...,cm−1)∈Rm−1.Then
associatedhubmatrix.Sharperboundswillbepresentedin
Section4. m(cid:15)−1 m(cid:9)−1(cid:16)(cid:16) (cid:16)(cid:16)2m(cid:15)−1
The well-known result on the eigenvalues of arrowhead det(λI−Q)=(λ−b) (λ−dk)− (cid:16)cj(cid:16) (λ−dk).
matrices is the Cauchy interlacing theorem for Hermitian k=1 j=1 k=1
matrices [12]. We assume that the diagonal elements dj, k=/ j
(6)
j = 1,2,...,m−1, of the diagonal matrix D in (1) satisfy
therelationd1 ≤ d2 ≤ ··· ≤ dm−1.Letλ1,λ2,...,λm bethe
eigenvalues of Q arranged in increasing order. The Cauchy The proof of this result can be found in [5, 13] and
thereforeisomittedhere.
interlacingtheoremsaysthat
WhenthediagonalmatrixD in(1)isazeromatrix,the
λ1≤d1≤λ2≤d2≤···≤dm−2≤λm−1≤dm−1≤λm. (2) followingresultisfollowedfromLemma1.
When the vector c and the real number b in (1) are taken Corollary2. LetQ ∈ Rm×m beanarrowheadmatrixhaving
intoconsideration,alowerboundofλ andanupperbound thefollowingform:
1
of λm were developed by using the well-known Gershgorin ⎡ ⎤
theorem(see,e.g.,[3,12]),thatis, 0 c
Q=⎣ ⎦, (7)
⎧ ⎫ ct b
⎨ m(cid:9)−1 ⎬
λm<max⎩d1+|c1|,...,dm−1+|cm−1|,b+ i=1|ci|⎭, (3) wherec isavectorinRm−1 andb isarealnumber.Thenthe
⎧ ⎫ eigenvaluesofQare
⎨ m(cid:9)−1 ⎬
λ1>min⎩d1−|c1|,...,dm−1−|cm−1|,b− |ci|⎭. (4) (cid:13) (cid:13)
i=1 b− b2+4(cid:4)c(cid:4)2 b+ b2+4(cid:4)c(cid:4)2
λ1(Q)= 2 , λm(Q)= 2 ,
Accurate bounds of eigenvalues of arrowhead matrices
are of great interest in applications as mentioned before. λi(Q)=0, fori=2,...,m−1.
The main results of this paper are presented in Theorems (8)
11and12fortheupperandlowerboundsofthearrowhead
matrices. It is also shown in Corollary13 that the resulting Proof. ByusingLemma1,wehave
boundsaretighterthanin(2),(3),and(4).
(cid:17) (cid:18)
Therestofthepaperisoutlinedasfollows.InSection2, det(λI−Q)=λm−2 λ2−bλ−(cid:4)c(cid:4)2 . (9)
wewillintroducenotationandpresentseveralusefulresults
on the eigenvalues of arrowhead matrices. We give our
main results in Section3. In Section4, we revisit the lower Clearly,λ=0isazeroofdet(λI−Q)withmultiplicitym−2.
and upper bounds of the ratio of eigenvalues of arrowhead The z(cid:19)eros of the quadratic poly(cid:19)nomial λ2 −bλ−(cid:4)c(cid:4)2 are
matrices associated with hub matrices and wireless com- (b − b2+4(cid:4)c(cid:4)2)/2 and (b + b2+4(cid:4)c(cid:4)2)/2, respectively.
munication systems [9], and subsequently, we make those Thiscompletestheproof.
boundsshaperbyusingtheresultsinSection3.InSection5,
we compute the bounds of arrowhead matrices using the Inwhatfollows,amatrixQhavingaformin(7)iscalled
developed theorems via three examples. Conclusions are
aspecialarrowheadmatrix.Thefollowingcorollary(also,see
giveninSection6.
[3])isadirectresultfromLemma1.
2.NotationandBasicResults Corollary3. LetQ beanm×marrowheadmatrixgivenby
(1),whereD =diag(d1,d2,...,dm−1)∈R(m−1)×(m−1),b∈R,
The identity matrix is denoted by I. The notation andc=(c1,c2,...,cm−1)∈Rm−1.Letusdenotetherepetition
diag(a1,a2,...,an)representsadiagonalmatrixwhosediag- ofthenumberdj inthesequence{di}im=−11bykj.Ifk j ≥2,then
onalelementsarea1,a2,...,an.Thedeterminantofamatrix dj istheeigenvalueofQwithmultiplicitykj−1.
A is denoted by det(A). The eigenvalues of a symmetric
matrixA∈Rn×narealwaysorderedsuchthat Proof. When the integer kj ≥ 2, the result follows from
Lemma1 since (λ − dj)kj−1 is a factor of the polynomial
λ1(A)≤λ2(A)≤···≤λn(A). (5) det(λI−Q).
For a v(cid:13)ec(cid:14)tor a ∈ Rn, its Euclidean norm is defined to be Corollary 4. Let Q be an m × m arrowhead matrix given
(cid:4)a(cid:4):= ni=1|ai|2. by (1), whereD = diag(d1,d2,...,dm−1) ∈ R(m−1)×(m−1),
Thefirstresultisaboutthedeterminantofanarrowhead b ∈ R, and c = (c1,c2,...,cm−1) ∈ Rm−1. Suppose that the
matrixandisstatedasfollows. lastk ≥ 2diagonalelementsdm−k,dm−k+1,...,dm−1 ofD are
2
EURASIPJournalonAdvancesinSignalProcessing 3
identicalanddistinctfromthefirstm−k−1diagonalelements entriesofthevectorcin(1)arenonzero.Thereasonforthis
d1,d2,...,dm−k−1ofD.Defineanewmatrix assumptionisthefollowing.Supposethatcj,the jthentryof
⎡ ⎤ c,isnonzero,itcanbeeasilyseenfromLemma1thatλ−dj
d1 c(cid:20)1 isafactorofdet(λI −Q);thatis,dj isoneofeigenvaluesof
⎢ ⎥
⎢⎢⎢⎢ ... ... ⎥⎥⎥⎥ Qa.mTahtreixrewmhaiicnhinigseoibgteanivnaelduebsyosfimQpalryedtheleetsianmgethaestjhthosreoowf
Q(cid:20) :=⎢⎢ d c(cid:20) ⎥⎥ (10) and column ofQ . In summary, for any arrowhead matrix,
⎢ m−k−1 m−k−1⎥
⎢ ⎥ wecanfindeigenvaluescorrespondingtorepeatedvaluesin
⎢⎣ dm−k c(cid:20)m−k ⎥⎦ Dorassociatedwithzeroelementsincbyinspection.
c(cid:20) ··· c(cid:20) c(cid:20) b In this paper, we call a matrix Q in (1) irreducible if
1 m−k−1 m−k
the diagonal elements d1,d2,...,dm−1 of Q are distinct and
w(cid:13)(cid:14)ith c(cid:20)j = cj for j = 1,2,...,m − k − 1 and c(cid:20)m−k = all elements of c are nonzero. By using Corollary4 and the
mj=−m1−k|cj|2.ThentheeigenvaluesofQarethatofQ(cid:20)together above discussion, this arrowhead matrix can be reduced to
withdm−k withmultiplicityk−1. anirreducibleone.
aPnrododf.isStiinnccetfrnoummnbuermsbdemrs−dk,1d,dm2−,k.+.1.,,.d.m.,−dkm−1−,1waerheavideentical c(R1oe)nmsisaidrakevree5dc.t;oItnhraint[4iCs,,mi9t−]a1,l.loWHweesrcmtahniattidacinrinecattrhlryeocwmohanetsratidrxuQcmtoamftratihnceeysf(oraremrael
⎛ ⎞
symmetric)arrowheadmatricesdenotedbyQ(cid:20) fromQ.The
m(cid:15)−1(λ−di)=⎜⎜⎜⎝m(cid:15)−k(λ−di)⎟⎟⎟⎠(λ−dm−k)k−1, j≤m−k−1, darieagtohneaelxealcetmlyesnatms eofasthtehsoesesyomfmQ.etTrihceavrercotworhec(cid:20)aidnmQ(cid:20)actoriucleds
i=1 i=1
i=/ j i=/ j bechosenas
m(cid:9)−1 (cid:16)(cid:16)(cid:16)cj(cid:16)(cid:16)(cid:16)2m(cid:15)−1(λ−di) c(cid:20)=(±|c1|,±|c2|,...,±|cm−1|). (13)
j=m−k i=1
i=/ j In such a way, there are 2m−1 such symmetric arrowhead
=⎛⎝ m(cid:9)−1 (cid:16)(cid:16)(cid:16)cj(cid:16)(cid:16)(cid:16)2⎞⎠⎛⎝m−(cid:15)k−1(λ−di)⎞⎠(λ−dm−k)k−1. mevaetrryicseusc.hBseycmaumseetdreict(aλrIr−owQh)ea=dmdeatt(rλixI −Q(cid:20)Qh(cid:20)a)sbtyhLeeimdemnatic1a,l
j=m−k i=1 eigenvalueswithQ.Thisisthereasonwhywejustconsider
(11) theeigenvaluesofrealarrowheadmatricesinthispaper.
By(6)inLemma1,wehave Thefollowingwell-knownresultbyWeyloneigenvalues
of a sum of two symmetric matrices is used in the proof of
det(λI−Q)
ourmaintheorem.
=(λ−b)m(cid:15)−1(λ−di)−m(cid:9)−1(cid:16)(cid:16)(cid:16)cj(cid:16)(cid:16)(cid:16)2m(cid:15)−1(λ−di) Tmhaetroirceesm.L6et(uWseayssl)u.mLeetthFatathnedeGigebnevatlwuoesmof×F,mG,saynmdmFe+trGic
i=1 j=1 i=1
i=/ j havebeenarrangedinincreasingorder.Then
⎛ ⎞
=⎜⎜⎜⎝(λ−b)m(cid:15)i=−1k(λ−di)−m(cid:9)j=−1k(cid:16)(cid:16)(cid:16)c(cid:20)j(cid:16)(cid:16)(cid:16)2m−(cid:15)i=k1−1(λ−di)⎟⎟⎟⎠(λ−dm−k)k−1 λλjj((FF++GG))≤≥λλii((FF))++λλjj−−ii++1m((GG)),, ffoorrii≤≥jj., ((1154))
i=/ j
(cid:17) (cid:18)
=det λI−Q(cid:20) ·(λ−dm−k)k−1. Proof. See[14,page62]or[12,page184].
(12)
To apply Theorem6 for estimating eigenvalues of an
Clearly, if λ is an eigenvalue of Q, then λ is either an irreducible arrowhead matrix Q, we need to decompose Q
eigenvalue of Q(cid:20) or dm−k. Conversely,d m−k is an eigenvalue intoasumoftwosymmetricmatriceswhoseeigenvaluesare
ofQwithmultiplicityk−1andtheeigenvaluesofQ(cid:20)arethat relatively easy to be computed. Motivated by the structure
ofQ.Thiscompletestheproof. ofthearrowheadmatrixandtheeigenstructureofaspecial
arrowheadmatrix(see,Corollary2),wewriteQintoasum
ByusingCorollaries3and4,tostudytheeigenvaluesof ofadiagonalmatrixandaspecialarrowheadmatrix.
Q,wemayassumethatthediagonalelementsd1,d2,...,dm−1 To be more precisely, let Q ∈ Rm×m be an irreducible
of Q are distinct when we study the eigenvalues of Q in arrowheadmatrixasfollows:
(1).Sinceeigenvaluesofsquarematricesareinvariantunder
⎡ ⎤
similaritytransformations,wecanwithoutlossofgenerality D c
arrange the diagonal elements to be ordered so that d < Q=⎣ ⎦, (16)
1 ct d
d2 < ··· < dm−1. Furthermore, we may assume that all m
3
4 EURASIPJournalonAdvancesinSignalProcessing
where dm ∈ R, D = diag(d1,d2,...,dm−1) with 0 ≤ d1 < for j =1,2,...,m.ByCorollary2,wehave
d2 <···<dm−1 ≤dm,andcisavectorinRm−1.Foragiven
ρ∈[0,1],wewrite λ1(S)=s, λm(S)=t, λj(S)=0, for j =2,...,m−1,
(23)
Q=E+S, (17)
wheresandtaregivenby(21).
where
⎡ ⎤
E=diag(cid:29)d1,d2,...,dm−1,ρdm(cid:30), S=⎣c0t (cid:29)1−cρ(cid:30)dm(1⎦8.) UpperBounds. Byλj((1Q4))≤inλTih(Eeo)r+emλm6+,j−wi(eSh)ave (24)
Therefore, we can use Theorem6 to give estimates of the foralli≥ j.Clearly,foragiven j,
eigenvalues of Q via those of E and S. To number the !
eigenvaluesofE,weintroducethefollowingdefinition. λj(Q)≤min λi(E)+λm+j−i(S) . (25)
i≥j
Definition7. Foranumberρ∈[0,1],wedefineanoperator
Tρ that maps a sequence {di}mj=1 satisfying 0 ≤ d1 < d2 < M0,oarnedptre≥ci0se,lwy,eshinacvee{d(cid:20)i}mi=1ismonotonicallyincreasing,s≤
··· < dm−1 ≤ dm toanewsequence{d(cid:20)i}mj=1 := Tρ({di}mj=1) !
accordingtothefollowingrules:ifρdm ≤d1,thend(cid:20)1 :=ρdm λ1(Q)≤min d(cid:20)1+t,d(cid:20)2,...,d(cid:20)m−1,d(cid:20)m+s
and d(cid:20)j+1 := dj for j = 1,...,m−1; if ρdm > dm−1, then !
d(cid:20)j := dj for j = 1,...,m−1 and d(cid:20)m := ρdm; otherwise, =min d(cid:20)1+t,d(cid:20)2,d(cid:20)m+s ,
dtj(cid:20)hj=e:r=ejed+xji1sft,os.r.a.n,jm=in−t1e,g1.e..r.,j0j0s,ud(cid:20)cjh0+1th:a=t dρjd0m<,aρnddmd(cid:20)≤j+1d:j0=+1,dtjhfeonr λj(Q)≤min d(cid:20)j+t,d(cid:20)j+1,...,d(cid:20)m!=min d(cid:20)j+t,d(cid:20)j+1!(26)
0
Theorem 8. Let Q ∈ Rm×m be an irreducible arrow- for j =2,...,m−1,and
head matrix having a form of (16), where D =
diag(d1,d2,...,dm−1)with0 ≤ d1 < d2 < ··· < dm−1 ≤ dm, λm(Q)≤λm(E)+λm(S)=d(cid:20)m+t. (27)
and c is a vector in Rm−1. For a given ρ ∈ [0,1], define
{d(cid:20)i}mj=1:=Tρ({di}mj=1).Then,onehas Inconclusion,(19)holds.
⎧ !
⎪⎪⎪⎪⎨min d(cid:20)1+t,d(cid:20)2,d(cid:20)!m+s , if j =1, LowerBounds. By(15)inTheorem6,wehave,foragiven j,
λj(Q)≤⎪⎪⎪⎪⎩md(cid:20)min+td(cid:20),j+t,d(cid:20)j+1 , iiff2j =≤mj ≤, m−1, λj(Q)≥mi≤ajx λi(E)+λj−i+1(S)!. (28)
(19) Hence,
⎧
⎪⎪⎪⎪⎨d(cid:20)1+ s, ! if j =1, λ1(Q)≥λ1(E)+λ1(S)=d(cid:20)1+s,
! !
λj(Q)≥⎪⎪⎪⎪⎩mmaaxx dd(cid:20)(cid:20)1j−+1,td,(cid:20)dj(cid:20)m+−s1,,d(cid:20)m+s!, iiff2j =≤mj ≤, m−1, λj(Q)≥max d(cid:20)j+s,d(cid:20)j−1,...,d(cid:20)1 =max d(cid:20)j+s,d(cid:20)j−1(29)
(20)
for j =2,...,m−1,and
where !
s= (cid:29)1−ρ(cid:30)dm−(cid:13)(cid:29)1−ρ(cid:30)2dm2 +4(cid:4)c(cid:4)2, λm(Q)≥max d(cid:20)m+s,d(cid:20)m−1,...,d(cid:20)2!,d(cid:20)1+t (30)
2 =max d(cid:20)m+s,d(cid:20)m−1,d(cid:20)1+t .
(cid:13) (21)
(cid:29) (cid:30) (cid:29) (cid:30)
t= 1−ρ dm+ 1−ρ 2dm2 +4(cid:4)c(cid:4)2.
As we can see from Theorem8, the lower and upper
2
boundsoftheeigenvaluesforQarefunctionsofρ∈[0,1]for
Proof. Foragivennumberρ ∈ [0,1],wesplitthematrixQ the given irreducible matrix Q. In other words, the bounds
into a sum of a diagonal matrix E and a special arrowhead ofeigenvaluesvarywiththenumberρ.Particularly,whenwe
matrixSaccordingto(17),whereEandSaredefinedby(18). chooseρ beingtheendingpoints,thatis,ρ = 0andρ = 1,
Clearly,weknowthat we can give an alternative proof of interlacing eigenvalues
theorem for arrowhead matrices (see, e.g., [12, page 186]).
λj(E)=d(cid:20)j (22) Thistheoremisstatedasfollows.
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EURASIPJournalonAdvancesinSignalProcessing 5
Theorem9(Interlacingeigenvaluestheorem). LetQ∈Rm×m Proof. InTheorem8,theupperboundsoftheeigenvaluesof
be an irreducible arrowhead matrix having a form in (16), Qin(19)aredeterminedbyd(cid:20)j, j = 1,2,...,m,ands andt
whereD = diag(d1,d2,...,dm−1)with0 ≤ d1 < d2 < ··· < in(21).Theycanbeviewedasfunctionsofρin[0,1].That
dm−1 ≤dm,andc isavectorinRm−1.LettheeigenvaluesofQ is,theupperboundsoftheeigenvaluesofQarefunctionsof
bedenotedby{λj}mj=1withλ1≤λ2≤···≤λm.Then ρintheinterval[0,1].Therefore,weareabletofindoptimal
bounds of the eigenvalues of Q by choosing proper ρ. The
λ1≤d1≤λ2≤d2≤···≤dm−2≤λm−1≤dm−1≤λm. upperboundsonλj(Q)for j =1,2≤ j ≤m−1,and j =m
(31) in(34)arediscussedseparately.
Proof. By using (19) withρ = 0 in Theorem8, we have UpperBoundofλ (Q). From(19),wehave
λj ≤ dj for j = 1,2,...,m−1. By using (20) withρ = 1 1 !
in Theorem8, we obtain λj ≥ dj−1 for j = 2,3,...,m. λ1(S)≤min d(cid:20)1+t,d(cid:20)2,d(cid:20)m+s , (35)
Combiningthesetwopartstogetheryieldsourresult.
where d(cid:20)k, s, and t are functions of ρ on the interval
The proof of the above result shows that we could [0,1]. In this case, we consider ρ in the following four
haveimprovedlowerandupperboundsforeacheigenvalue subintervals: [0,d1/dm], [d1/dm,d2/dm], [d2/dm,dm−1/dm],
opnfaexraatnmseiecrttreierodnρu.ciinbl[e0,a1r]r.owOhueramdaminatrreisxubltys wfinildlibneggaivnenopintimthael ad(cid:20)n1d+[tdm=−1/df4m(ρ,1),],d(cid:20)r2es=pecdti1v,ealyn.dFod(cid:20)rmρ+∈s [=0,df11/(dρm).],Fwoer hρav∈e
[d1/dm,d2/dm],wehaved(cid:20)1+t = d1+ f2(ρ),d(cid:20)2 = ρdm,and
3.MainResults d(cid:20)m +s = dm−1 + f1(ρ). For ρ ∈ [d2/dm,dm−1/dm], we have
d(cid:20)1+t =d1+ f2(ρ),d(cid:20)2 =d2,andd(cid:20)m+s=dm−1+ f1(ρ).For
Associated with the arrowhead matrix Q in Theorem8, we ρ ∈ [dm−1/dm,1],wehaved(cid:20)1+t = d1+ f2(ρ),d(cid:20)2 = d2,and
definefourfunctions fi,i=1,2,3,4,ontheinterval[0,1]as d(cid:20)m+s= f3(ρ).Hence
follow:
" (cid:13) # mind(cid:20) =d ,
f1(cid:29)ρ(cid:30):= 12 (cid:29)1−ρ(cid:30)dm− (cid:29)1−ρ(cid:30)2dm2 +4(cid:4)c(cid:4)2 , ⎧ ρ∈[0,1"] 2 # 1 & ’
Obvfffi234o(cid:29)(cid:29)(cid:29)uρρρs(cid:30)(cid:30)(cid:30)ly:::,===ρρ12dd"mm(cid:29)1++s−=ff12ρ(cid:29)(cid:29)f(cid:30)ρρ(cid:29)d(cid:30)(cid:30)ρm,.(cid:30)+, (cid:13)(cid:29)1t−=ρ(cid:30)f2(cid:29)dρm2(cid:30)+, 4(cid:4)c(cid:4)2#, ((3323)) mρ∈iVn(cid:17)d(cid:20)m+s(cid:18)=⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ddddmmmm−−−−1111++++ ffff1111"""ddddddddmmmm12mm−−#11,,##,, iiiiffffVVVV ====&&&0ddddd,dmmm12dmd−,,m11ddd,dmm21,m−’’1,,’,
1 2
⎧ & ’
twiniohtnehTrseehfpesi,rafiono=oldlfo1tow,fa2iron,eu3ggr,io4mvb,eisanseinrsbivymrae(tpsi2oul1enl)t,s.ba.buotuqtuimteounsoetfounliacsitwyeowfiflulnsece- (cid:17) (cid:18) ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨df21(+0),f2"dd2#, iiffVV ==&0dd,1dd,m1dd2,’,
Lf3emaTnmdheaf4p10rao.reoTfinhocefrfetuhanissicntleigomonnmstfah1eiasinnodtmerfi2vttabelod[t.0h,a1r]e.decreasingwhile mρ∈iVn d(cid:20)1+t =⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩dd1++ ff2"(1d)dmm,m−1#, iiffVV ==&&dddmmm2−,1d,dmm1m−’1.’,
1 2 d
m
Theorem 11. Let Q be an irreducible arrowhead matrix (36)
defined by (16) and satisfying all assumptions in Theorem8.
ThentheeigenvaluesofQareboundedaboveby Since 0 > f1(d1/dm) > f1(d2/dm) > f1(dm−1/dm), f2(0) ≥
⎧ $ " #% dm>d1,and f2(d2/dm)> f2(dm−1/dm)> f2(1)>0,wehave
⎪⎪⎪⎪⎪⎪⎨min d1,dm−1+ f1 ddmm−1 , if j =1, λ1(Q)≤min$d1,dm−1+ f1"ddm−1#%. (37)
λj(Q)≤⎪⎪⎪⎪⎪⎪⎩ddmj,−1+ f2"ddm−1#, iiff2j =≤mj ≤. m−1, UpperBoundofλj(Q),for2≤ j ≤m−1. F!rmom(19),wehave
m
(34) λj(Q)≤min d(cid:20)j+t,d(cid:20)j+1 . (38)
5
6 EURASIPJournalonAdvancesinSignalProcessing
Inthiscase,weconsiderρlyinginthefollowingfoursubin- Theorem 12. Let Q be an irreducible arrowhead matrix
tervals:[0,dj−1/dm],[dj−1/dm,dj/dm],[dj/dm,dj+1/dm],and defined by (16) and satisfying all assumptions in Theorem8.
[dj+1/dm,1],respectively.Forρ ∈[0,dj−1/dm],wehaved(cid:20)j + ThentheeigenvaluesofQareboundedbelowby
wH[thda=eevjn+ehd1c/adej(cid:20)d−vjm1e+,+d1(cid:20)tj]f=,2+(wρfte)4(ahρ=na)dvaednd(cid:20)jddj(cid:20)++j1d(cid:20)+j=+f2t1(dρ==j).dFdaojnj.rdF+ρod(cid:20)r∈f2j+ρ([1ρd∈)j=−a[1nd/dρjd/dmdd(cid:20)mm,jd+.,1djF/jod=+rm1/]dρd,jmw+∈1]e., λj(Q)≥⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ddm1a++x,ff1d""j−ddd1m11,d##j,,+ f1(ddmj )-, iiifff2jj ==≤1mj,≤. m−1,
1 2 d
m
mind(cid:20)j+1=dj, (45)
ρ∈[0,1]
Proof. In Theorem8, the lower bounds of the eigenvalues
⎧ ( ) * +
(cid:17) (cid:18) ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ddjj−−11++ ff22(ddddjjmm−−11),, iiffVV ==*0dd,jm−dd1jm−,1ddmj,+, [eooa0inffg,deQt1hn]tev.iianTneluhi(g(ee2e2sr0n1e)ofv)o.afarlArueQe,seswdwbeeyoetfeacdrrQhmiedoaioanibrnseleiednTgftuhboneypfiocrdornt(cid:20)iepjdom,enorjs1pρ2=ot.i,fmTt1ρhah,el2einb,lod.ow.tuih.s,encermudsisb,nsooitaouefnnrntdvhdsaiessl
mρ∈iVn d(cid:20)j+t =⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ddjj++ ff22((1d)d,jm+1), iiffVV ==**ddddmjj−,1d,d1jm++1.+, gLiovwenerfBoroujn=d1of,2λ1≤(Qj).≤λF1m(roQ−m)1≥(,2ad0(cid:20)n1)d,+wjse=.hmavein(45),separa(t4el6y).
m
(39) In this case, we consider ρ lying in the following two
subintervals: [0,d1/dm] and [d1/dm,1]. For ρ ∈ [0,d1/dm],
Therefore, d(cid:20)1+s= f3(ρ).Forρ∈[d1/dm,1],wehaved(cid:20)1+s=d1+f1(ρ).
Hence
, ( ) -
λj(Q)≤min dj−1+ f2 ddjm−1 ,dj . (40) (cid:17) (cid:18) ⎧⎪⎪⎪⎪⎨d1+ f1"ddm1#, ifV =&0,ddm1’,
Sincedj−1+ f2(dj−1/dm) = f4(dj−1/dm) > f4(0) ≥ dm ≥ dj, mρ∈aVx d(cid:20)1+s =⎪⎪⎪⎪⎩d1+ f1"dd1#, ifV =&dd1,1’. (47)
weget m m
Itleadsto
λj(Q)≤dj. (41) "d #
λ (Q)≥d + f 1 . (48)
1 1 1 d
m
UpperBoundofλm(Q). From(19)wehave
LowerBoundofλ (Q). From(20),wehave
2
λm(Q)≤d(cid:20)m+t. (42) !
λ (Q)≥max d(cid:20),d(cid:20) +s . (49)
2 1 2
Forρ ∈ [0,dm−1/dm],wehaved(cid:20)m+t = dm−1+ f2(ρ)while In this case, we consider ρ lying in the following three
forρ∈[dm−1/dm,1],wehaved(cid:20)m+t= f4(ρ): subintervals: [0,d1/dm], [d1/dm,d2/dm], and [d2/dm,1]. For
⎧ " # & ’ ρ ∈ [0,d1/dm],wehaved(cid:20)1 = ρdm,d(cid:20)2+s = d1+ f1(ρ).For
mρ∈iVn(cid:17)d(cid:20)m+t(cid:18)=⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ddmm−−11++ ff22"ddddmmmm−−11#,, iiffVV ==&0d,dmdm−dm1m−,11’,. ρρHe∈∈nc[[edd,21//ddmm,,1d]2,/dwme],hwamveeahxd(cid:20)ad1(cid:20)v1e==d(cid:20)1dd11=,andd1,d(cid:20)d2(cid:20)2++ss==df23(+ρ)f.1(Fρo)r.
(43) ρ∈V
⎧ & ’
Hence, "d # (cid:17) (cid:18) ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨d1+ f1("0d), # ifV =&0d,ddm1d,’ (50)
Thiscompletestλhme(pQro)o≤f.dm−1+ f2 dmm−1 . (44) mρ∈aVx d(cid:20)2+s =⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩dd2++ ff1"ddm22#,, iiffVV ==&ddm12,,1d’m2. ,
2 1 d d
m m
6
EURASIPJournalonAdvancesinSignalProcessing 7
Theseleadto ρ∈[dm−1/dm,1],wehaved(cid:20)1+t =d1+ f2(ρ),d(cid:20)m−1 =dm−1,
d(cid:20)m+s= f3(ρ).Hence
$ " #%
d
λ (Q)≥max d ,d + f 2 . (51)
2 1 2 1 dm max d(cid:20)m−1=dm−1,
ρ∈[0,1]
⎧ & ’
LowerBoundofλλjj((QQ)),3≥≤mjax≤ dm(cid:20)j−−1,1d(cid:20).j+Frso!m. (20),weha(v5e2) (cid:17) (cid:18) ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ddmm−−11++ ff11("0dd),1#, iiffVV==&0dd,1dd,m1ddm,−2’,
m m m
aItaennnrddtvha[idl(cid:20)sdsj:cj/+a[ds0mes,,,dw1=j]−e.2c/Fdodojnm−rs1]iρ,d+e[∈rdfρ1j−[(l0ρ2y,/)idd.nmjgF−,o2idn/rdj−tρmh1]/e,d∈fmwol]el[,odhw[ja−divn2je−/gd1dt/(cid:20)mhdj−,rmd1e,ej−d=s1ju//dddbmmijn−]]2-,, mρ∈aVx d(cid:20)m+s =⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ddmm−+1+f1(f11"),ddmm−2#, iiffVV==&&ddddmmm−−21,,d1dm’m,−1’,
m
w[ρHdee∈jn−h1c[a/edvdjem/d,d(cid:20)mdj−,j/11d]m,=w],eρwdhemahvaeavnde(cid:20)djd−(cid:20)d1j(cid:20)−j=1+=dsjd−=j1−a1ndadjn−dd1(cid:20)jd+(cid:20)+j+fs1s(=ρ=)d.fjF3+(oρr)f1.ρ(Fρo∈)r. (cid:17) (cid:18) ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨dd1++ ff2""dddm11##,, iiffVV ==&&0d,1dd,m1d’m,−2’,
1 2 d d d
max d(cid:20) +t = m m m
mρ∈aVx(cid:17)d(cid:20)j+s(cid:18)=⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ddddjjjj−−++11mρ++ff∈11aV((xff11ddd((dd(cid:20)0mjjj−d)d)),1jm−,,=2)d,j−1iiii,ffffVVVV ====****0dddddd,jjjmm−−dd,21j1m−,,+2dddd+.mjjm−,+1+, , Sf2iρn(∈dcVλemm−0(2Q>/1d)mf1≥)(0>m)fa>2x⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩($dddf1md11(m−d++1−1//1ffdd,22md""mm))dd,ddmm+>wmm−−e21ff11h##((a1,,dv)me,−diiff21/VV+dm==f)2"a&&ndddddmdd1mmmm−−#f221%(,,d.1d1’dm/.md−m1((55)’76,>))
m m
(53) Sinced1+ f2(d1/dm)= f4(d1/dm)> f4(0)≥dm,weget
" #
d
Sincedj−1>dj−1+ f1(0)>dj−1+ f1(dj−2/dm),wehave λm(Q)≥d1+ f2 d1 . (58)
m
, ( )-
λj(Q)≥max dj−1,dj+ f1 ddj . (54) Thiscompletestheproof.
m
Corollary 13. Let Q be an irreducible arrowhead matrix
defined by (16) and satisfying all assumption in Theorem8.
LowerBoundofλm(Q). From(20),wehave ThenupperandlowerboundsoftheeigenvaluesofQobtained
byTheorems11and12aretighterthanthosegivenby(2),(3),
! and(4).
λm(Q)≥max d(cid:20)1+t,d(cid:20)m−1,d(cid:20)m+s . (55)
Proof. Since
Inthiscase,weconsiderρlyinginthefollowingthreesubin- $ " #%
d
tervals: [0,d1/dm], [d1/dm,dm−2/dm], [dm−2/dm,dm−1/dm], min d1,dm−1+ f1 dm−1 ≤d1, (59)
and [dm−1/dm,1]. For ρ ∈ [0,d1/dm], we have d(cid:20)1 + t = m
f4(ρ), d(cid:20)m−1 = dm−2, d(cid:20)m + s = dm−1 + f1(ρ). For ρ ∈ thentheupperboundfortheeigenvalueλ (Q)givenby(34)
[d1/dm,dm−2/dm],wehaved(cid:20)1+t=d1+ f2(ρ),d(cid:20)m−1 =dm−2, inTheorem11istighterthanthatby(2).T1heupperbounds
d(cid:20)m+s=dm−1+ f1(ρ).Forρ∈[dm−2/dm,dm−1/dm],wehave fortheeigenvaluesλj(Q), j =2,...,m−1,providedby(34)
d(cid:20)1+t =d1+ f2(ρ),d(cid:20)m−1 =ρdm,d(cid:20)m+s=dm−1+ f1(ρ).For inTheorem11arethesameasthoseby(2).
7
8 EURASIPJournalonAdvancesinSignalProcessing
Notethat0≤d1<···<dm−1≤dm;theright-handside 4.HubMatrices
of (3)withb =dmis
Using the improved upper and lower bounds for the
⎧ ⎫
⎨ m(cid:9)−1 ⎬ arrowheadmatrix,wewillnowexaminetheirapplicationsto
max⎩d1+|c1|,...,dm−1+|cm−1|,b+ |ci|⎭ hub matrices and MIMO wireless communication systems.
i=1 Theconceptofthehubmatrixwasintroducedinthecontext
(60)
ofwirelesscommunicationsbyKungandSuterin[9]andit
m(cid:9)−1
=dm+ |ci|. isreexaminedhere.
i=1 Definition17. AmatrixA∈Rn×miscalledahubmatrix,ifits
Sf2in(dcem−(cid:4)1c/(cid:4)dm)≤=(cid:14)f4im(=d−1m1−|c1i/|d,md)m,w+eh(cid:4)ac(cid:4)ve = f4(1), and dm−1 + tfiorsetamch−o1thcoerluwminths(rceaslpleedctntoonhthuebEcoulculimdenasn)ainrenoerrthporogdouncatl
and its last column (called hub column) has its Euclidean
m(cid:9)−1 & "d #’ "d # normgreaterthanorequaltothatofanyothercolumns.We
dm+ |ci|− dm−1+ f2 dm−1 ≥ f4(1)− f4 dm−1 >0, assumethatallcolumnsofAarenonzerosvectors.
i=1 m m
(61) We denote the columns of a hub matrix A by
a1,a2,...,am.Vectorsa1,a2,...,am−1areorthogonaltoeach
andthentheupperboundofλm(Q)from(34)inTheorem11 other. We further assume that 0 < (cid:4)a1(cid:4) ≤ (cid:4)a2(cid:4) ≤ ··· ≤
istighterthanthatfrom(3). (cid:4)am(cid:4). In such case, we callA an ordered hub matrix. Our
Nowweturntothelowerboundsofλj(Q).Since interest is to study the eigenvalues ofQ = AtA, the Gram
matrixA.Inthecontextofwirelesscommunicationsystems,
, ( )-
d Qisalsocalledthesystemmatrix.ThematrixQhasaform
max dj−1,dj+ f1 dj ≥dj−1 (62) asfollows:
m
⎡ ⎤
for j =2,...,m−1and ⎢ (cid:4)a1(cid:4)2 (cid:6)a1,am(cid:7) ⎥
⎢ ⎥
"d # ⎢⎢ (cid:4)a2(cid:4)2 (cid:6)a2,am(cid:7) ⎥⎥
d1+ f2 dm1 ≥dm>dm−1, (63) Q=⎢⎢⎢⎢ ... ... ⎥⎥⎥⎥.
⎢ ⎥
⎢ ⎥
we know that the lower bounds for the eigenvalues λj(Q), ⎢⎣ (cid:4)am−1(cid:4)2 (cid:6)am−1,am(cid:7)⎥⎦
tjha=nt2h,o.s.e.,bmy,(2p)r.ovided by (45) inTheorem12 are tighter (cid:6)am,a1(cid:7) (cid:6)am,a2(cid:7) ··· (cid:6)am,am−1(cid:7) (cid:4)am(cid:4)2
(66)
Remark 14. When c in (16) is a zero vector, by using
Theorems 11 and 12, we haved j ≤ λ(Q) ≤ dj, that is, Clearly,QisanarrowheadmatrixassociatedwithA.
λ(Q) = dj.Inthissense,thelowerandupperboundsgiven An important way to characterize properties of Q is in
inTheorems11and12aresharp. termsofratiosofitssuccessiveeigenvalues.Tothisend,the
ratiosarecalledeigengapofQwhicharedefined[9]tobe
Remark 15. When Q in Theorems 11 and 12 has size of
i2d×en2ti,ctahle.Aucptpuearllay,nfdrolomwTerhbeooruenmdss1o1fiatnsdea1c2hweeighenavvealueare EGi(Q)= λmλ−m(−i−i(1)Q(Q) ) (67)
" # $ " #%
d d
d + f 1 ≤λ (Q)≤min d ,d + f 1 , for i = 1,2,...,m−1. Following the definition in [9], we
1 1 d 1 1 1 1 d
2 2 definetheithhub-gapofAasfollows:
" # " # (64)
d d
d + f 1 ≤λ (Q)≤d + f 1 . . .
1 2 d 2 1 2 d .a .2
2 2 HGi(A)= (cid:4)ma−(i−(cid:4)1)2 (68)
m−i
Clearly,wehave
"d # "d # fori=1,2,...,m−1.
λ (Q)=d + f 1 , λ (Q)=d + f 1 . (65) The hub-gaps of A will allow us to predict the eigen-
1 1 1 d 2 1 2 d
2 2 structureofQ.Itwasshownin[9]thatthelowerandupper
ThiscanbeverifiedbycalculatingtheeigenvaluesQdirectly. boundsofEG1(Q)[9]aregivenbythefollowing:
Remark 16. For the lower bound of the smallest eigenvalue HG1(A)≤EG1(Q)≤(HG1(A)+1)HG2(A). (69)
ofanarrowheadmatrix,noconclusioncanbemadeforthe
tightnessoftheboundsbyusing(4)and(45)inTheorem12. Theseboundsonlyinvolvenonhubcolumnshavingthetwo
Anexamplewillbegivenlater(seeExample22inSection5). largest Euclidean norms and the hub column of A. Using
8
EURASIPJournalonAdvancesinSignalProcessing 9
the results in Theorems 11 and 12, we obtain the following Proof. Wefirstneedtoshow
bounds: (cid:17) (cid:18)
f (cid:17)(cid:4)a (cid:4)2/(cid:4)a (cid:4)2(cid:18) (cid:4)am(cid:4)2 < f4 (cid:4)a1(cid:4)2/(cid:4)am(cid:4)2 . (76)
4 1 m (cid:4)a (cid:4)2 (cid:4)a (cid:4)2
m−1 m−1
(cid:4)a (cid:4)2
m−1
(cid:17) (cid:18) Clearly,thisistruebecauseof (71).Nextweneedtoshow
f (cid:4)a (cid:4)2/(cid:4)a (cid:4)2
≤EG (Q)≤ 4 m−1(cid:17) m (cid:18)!. (cid:17) (cid:18)
1 max (cid:4)am−2(cid:4)2, f3 (cid:4)am−1(cid:4)2/(cid:4)am(cid:4)2 f4 (cid:4)am−1(cid:17)(cid:4)2/(cid:4)am(cid:4)2 (cid:18)!
(70) max (cid:4)am−2(cid:4)2, f3 (cid:4)am−1(cid:4)2/(cid:4)am(cid:4)2
(77)
( )
Obviously,theseboundsarenotonlyrelatedtotwononhub (cid:4)a (cid:4)2 (cid:4)a (cid:4)2
< m +1 m−1 .
columns with the largest Euclidean norms and the hub (cid:4)a (cid:4)2 (cid:4)a (cid:4)2
columnofAbutalsorelatedtothenonhubcolumnhaving m−1 m−2
the smallest Euclidean norm and interrelationship between Tothisend,itissufficetoprove
all nonhub columns and the hub column of A. As we
( )
eshxpouecldtebde, ttihgehtleorwtehrananthdouseppiner(6b9o)u.TnodsproofveEGth1i(sQst)atienm(e7n0t), f4 (cid:4)(cid:4)aam−(cid:4)1(cid:4)22 <(cid:4)am(cid:4)2+(cid:4)am−1(cid:4)2. (78)
m
wegivethefollowinglemmafirst.
Thisisexactly(72).Theproofiscomplete.
Lemma18. Leta1,a2,...,am bethecolumnsofahubmatrix
Awith0<(cid:4)a1(cid:4)≤(cid:4)a2(cid:4)≤···≤(cid:4)am−1(cid:4)≤(cid:4)am(cid:4).Then Thelowerboundin(70)canberewrittenintermsofthe
f4(cid:29)ρ(cid:30)>(cid:4)am(cid:4)2 forρ∈(0,1], (71) hubgapofAasfollows:
( ) (cid:17) (cid:18) ⎡ 1 ⎤
2
f4 (cid:4)(cid:4)aamm−(cid:4)1(cid:4)22 <(cid:4)am(cid:4)2+(cid:4)am−1(cid:4)2. (72) f4 (cid:4)(cid:4)a1a(cid:4)m2−/1(cid:4)(cid:4)a2m(cid:4)2 = 12HG1(Q)⎢⎣1+231+ (cid:4)4a(cid:4)mc(cid:4)(cid:4)24⎥⎦. (79)
Proof. FromLemma10,weknow,forρ∈(0,1],
The upper bound in (70) can be rewritten in terms of the
(cid:13) hubgapofAasfollows:
f4(cid:29)ρ(cid:30)> f4(0)= (cid:4)am(cid:4)2+ (cid:4)a2m(cid:4)4+4(cid:4)c(cid:4)2 >(cid:4)am(cid:4)2, (73) f (cid:17)(cid:4)a (cid:4)2/(cid:4)a (cid:4)2(cid:18)
4 m−1(cid:17) m (cid:18)!
(cid:14)
where(cid:4)c(cid:4)2= im=−11|(cid:6)ai,am(cid:7)|2.Theinequality(71)holds.By max (cid:4)am−2(cid:4)2, f3 (cid:4)am−1(cid:4)2/(cid:4)am(cid:4)2
thedefinitionof f ,showingtheinequality(72)isequivalent (cid:17) (cid:18)
4
toproving f4 (cid:4)am−1(cid:4)2/(cid:4)am(cid:4)2
≤
(cid:4)a (cid:4)2
(cid:4)c(cid:4)2≤(cid:4)am(cid:4)2(cid:4)am−1(cid:4)2. (74) m−2
1
= (HG (A)+1)HG (A)
Thisistruebecause 2 1 2
1
2
(cid:4)am(cid:4)2≥m(cid:9)j=−11...a1j...2(cid:16)(cid:16)(cid:16)/aj,am0(cid:16)(cid:16)(cid:16)2 + 12(HG1(A)−1)HG2(A)2231+ (cid:17)(cid:4)am(cid:4)24−(cid:4)c(cid:4)(cid:4)a2m−1(cid:4)2(cid:18)2.
(75) (80)
m(cid:9)−1 1 (cid:16)(cid:16)/ 0(cid:16)(cid:16)2 (cid:4)c(cid:4)2
≥ j=1(cid:4)am−1(cid:4)2(cid:16) aj,am (cid:16) = (cid:4)am−1(cid:4)2. TocomparetheseboundstoKungandSuter[9],set(cid:4)c(cid:4)2 =
0,andtheboundsforEigGap (Q)in(70)become
1
Thefirstinequalityofaboveisfromtheorthogonalityofaj,
j =1,...,m−1whilethesecondinequalityisfrom(cid:4)a (cid:4)≤ HG1(A)≤EG1(Q)≤HG1(A)HG2(A). (81)
1
(cid:4)a2(cid:4)≤···≤(cid:4)am−1(cid:4).Thiscompletestheproof.
Under these conditions, the lower bound agrees with Kung
andSuterwhiletheupperboundistighter.
Thefollowingresultholds.
Let A ∈ Rn×m be an ordered hub matrix. Let A(cid:20) ∈
Proposition 19. Let Q in (66) be the arrowhead matrix Rn×k be a hub matrix obtained by removing the first n−k
associated with a hub matrix A. Assume that 0 < (cid:4)a (cid:4) < nonhub columns of A with the smallest Euclidean norms.
1
(cid:4)a2(cid:4) < ··· < (cid:4)am−1(cid:4) ≤ (cid:4)am(cid:4), where aj, j = 1,...,m This corresponds to the MIMO beamforming problem by
arecolumnsofA.ThentheboundsoftheEG (Q)in(70)are comparing k of m transmitting antennas with the largest
1
tighterthanthosein(69). signal-to-noise ratio (see [9]). The ratio λk(Q(cid:20))/λm(Q) with
9