Maximal doubly stochastic matrix centralizers

Contents lists available atScienceDirect

Linear

Algebra

and

its

Applications

www.elsevier.com/locate/laa

Maximal

doubly

stochastic

matrix

centralizers

HenriqueF. da Cruza,Gregor Dolinarb,c, Rosário Fernandesd,∗,

Bojan Kuzmae,c

a

DepartamentodeMatemáticadaUniversidadedaBeiraInterior,RuaMarquês D’AvilaeBolama,6201-001Covilhã,Portugal

b_{UniversityofLjubljana,FacultyofElectricalEngineering,Tržaškacesta25,}

1000 Ljubljana,Slovenia

c_{IMFM,Jadranskacesta19,1000 Ljubljana,Slovenia}

d_{DepartamentodeMatemática,FaculdadedeCiênciaseTecnologia,Universidade}

NovadeLisboa,2829-516Caparica,Portugal

e

UniversityofPrimorska,Glagoljaška8,6000 Koper,Slovenia

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received21September2016 Accepted19June2017 Availableonline3July2017 SubmittedbyM.Tsatsomeros

MSC:

15A27

Keywords:

Doublystochasticmatrices Matrixalgebra

Centralizers

Wedescribedoublystochasticmatriceswithmaximal central-izers.

©2017ElsevierInc.Allrightsreserved.

✩ _{ThisworkwaspartiallysupportedbyPortuguesefundsofFCT—FoundationforScienceandTechnology}

undertheprojectsUID/MAT/00212/2013andUID/MAT/00297/2013,andbySlovenianResearchAgency (researchcorefundingNo. P1-0288andNo. P1-0222).

* Correspondingauthor.

E-mailaddresses:[email protected](H.F. daCruz),[email protected](G. Dolinar),

mrff@fct.unl.pt(R. Fernandes),[email protected](B. Kuzma).

http://dx.doi.org/10.1016/j.laa.2017.06.029

1. Introduction

LetMn bethealgebraofalln-by-n matricesoverthecomplexfieldC andletI∈ Mn

beitsidentity.OneoftherelationswhichisoftenusedonMn,bothinpureandapplied

problems,iscommutativity[13–16].Inthestudyofcommutativitythenotionof central-izer (alsocalledcommutant)hasanimportantrole.ForA∈ Mn itscentralizer,denoted

byC(A),isthesetof allmatrices commutingwith A,thatis

C(A) = {X ∈ Mn : AX = XA},

and foraset S⊆ Mn itscentralizer, denotedbyC(S),istheintersectionof centralizers

of allitselements,thatis

C(S) = {X ∈ Mn: AX = XA, for every A∈ S}.

TheCentralizerTheorem(see[6],p. 113,Corollary 1,or[16],p. 106,Theorem 2)states thatC(C(A))=C[A] whereC[X]⊆ Mn denotestheunitalalgebraspannedbyX∈ Mn.

ItiswellknownthatthecentralelementsofMnarethescalarmatrices,C(Mn)={αI :

α∈ C}.

The centralizer induces an equivalence relation, ∼, and a preorder relation, , on

Mn:

• A andB are C-equivalent,A∼ B,ifC(A)=C(B),

• A B ifC(A)⊆ C(B).

Forapreorder onMn wesaythat:

• anon-scalarmatrixA isminimalifforeverymatrixX withC(X)⊆ C(A) itfollows

C(A)=C(X),

• anon-scalarmatrixA ismaximalifforeverynon-scalarmatrixX withC(X)⊇ C(A)

itfollowsC(A)=C(X).

Thecharacterizationofminimal andmaximalmatricesinMn isknown.Forminimal

matrices theclassificationconsistsofseveral equivalentconditions,welistonlyafewof them.

Proposition 1.1(see [4]).ForA∈ Mn thefollowingstatements are equivalent.

(1) A is minimal.

(2) A is nonderogatory,i.e.,theminimalpolynomialof A equalsitscharacteristic poly-nomial.

Formaximalmatrices,thecharacterizationisasfollows:

Proposition 1.2(see Lemma 3.1 in [9]). Amatrix A∈ Mn ismaximal if and only if it

is C-equivalent to a non-scalar idempotent or C-equivalent to a non-scalar square-zero matrix.

Itisouraimto classifydoublystochasticmatrices,denotedhenceforthbyΩn⊆ Mn,

withextremal centralizers.Recall thatadoubly stochasticmatrix is asquarematrix of nonnegativerealnumberswitheachrowandcolumnsummingto1.Itiswell-known(see, e.g.[8, Theorem A.2])thatΩn isaconvexclosureoftheset ofpermutationalmatrices.

DoublystochasticmatricesareimportantinthestudyofMarkovchains,combinatorics, statisticsandprobability[1,7,10,11].Matricesthatcommutewithdoublystochastic ma-trices are the subject of several papers, see for example [2] and [12]. In particular, in graphtheoryagraphG withadjacencymatrixA iscalledcompact ifA commuteswith everydoublystochasticmatrix.

Themainresultsofourpaperisacompleteclassificationofmaximaldoublystochastic matricesandmatriceswhicharemaximalwhentheircentralizersarerestrictedtodoubly stochastic matrices. Unlikefor maximal matrices,the set ofminimal doubly stochastic matricesisopen (seeRemark 3.8)and consistsofnonderogatorydoublystochastic ma-trices.Thereforenofurtherclassificationofminimaldoublystochasticmatricesisgiven. 2. Matriceswithmaximalcentralizers

Webeginthissectionwithacharacterizationofthemaximaldoublystochastic matri-ces.RecallthatinadoublystochasticmatrixP everyentryliesintheinterval[0,1]⊆ R. Also, it is a trivial observation that an entry-wise nonnegative matrix M is doubly-stochasticifandonlyifM 1= 1 and1T_{M = 1}T_{,where1= [1. . . 1]}T _{isacolumnvector}

ofonesand XT _{denotesatranspositionof(possiblynon-square)matrixX.}

Theorem2.1. Anon-scalarmatrixM is maximal doublystochastic ifandonly if M = I− α(I − P ),

where P = [pij] = I is a doubly stochastic idempotent and α ∈ R satisfies 0 < α ≤

min  1 1−pii : pii= 1  . Proof. If M = I− α(I − P ),

withP anidempotentdoublystochasticmatrixandα asinthestatementofthetheorem, then M is C-equivalent to a non-scalar idempotent and so by Proposition 1.2, M is

Moreover, P isdoubly stochastic so pii ≤ 1 and then theassumption on α gives that

diagonal entries of M are also nonnegative.Since clearly M 1 = 1 and1TM = 1T we see thatM is doublystochastic.

Conversely,suppose thatM isamaximaldoublystochasticmatrix.Then by Propo-sition 1.2,M isC-equivalenttoeitherasquare-zeromatrixor anidempotent matrix.

In thefirst case M = λI + N ,where N is asquare-zero matrix.Since M is doubly stochastic,

1 = M 1 = (λI + N )1 = λ1 + N 1. Then

N 1 = (1− λ)1

and as N issquare-zeroweconcludethatλ= 1 andM = I + N .Also, Tr(N )= 0 and eachdiagonalentryofN = M−I isnon-positivesinceM isdoublystochastic.Therefore eachdiagonalentryofN equalszeroandeachdiagonalentryofM = N + I equalsone. Since M isdoublystochasticthisgivesM = I,a contradiction.

Inthesecond caseM = λI + μQ with μ= 0 andQ anon-scalaridempotentmatrix. Clearly,thespectrumofM equals{λ,λ+ μ}.SinceM isdoublystochastic,1 is alsoits eigenvalue.Hence

λ = 1 or λ + μ = 1. (1)

Inaddition, sinceM ≥ 0 entry-wise,

0≤ Tr(M) = λ Tr(I) + μ Tr(Q) = λ n + μ rank(Q). (2)

Then, (1)–(2) implyλ,μ∈ R.BythePerron–Frobeniustheorem fornonnegative matri-ces, thespectral radius ρ(M ) isaneigenvalueand thecorresponding eigenvectorx can

be takenentrywisenonnegative.MultiplyingM x= ρ(M )x by1T _{ontheleft, weget}

1Tx = 1TM x = ρ(M )1Tx

andhenceρ(M )= 1. Therefore,as thespectrumofM equals{λ,λ+ μ}⊆ R,weobtain thefollowing:ifλ= 1 thenμ< 0 andifλ+ μ= 1 thenμ> 0.So,

M = I− α(I − P ),

where P = Q andα = μ > 0 ifλ+ μ= 1,and where P = I− Q and α =−μ> 0 if λ= 1.InbothcasesP isanidempotentmatrix.WearegoingtoprovethatP isinboth casesadoublystochastic matrix.

Since M = I− α(I − P ) is doublystochasticand α > 0,theoff-diagonals entriesof

pii= p2ii+ ε, where ε =



1≤k≤n k=i

pikpki≥ 0

andweconcludethatp2

ii≤ pii.Consequently,0≤ pii≤ 1.Moreover,

1 = M 1 = (I− α(I − P ))1

easilyimpliesP 1= 1 andlikewise1T_{M = 1}T _implies1T_{P = 1}T _{soP isdoubly}

stochas-tic.The conditionthat0< α ≤ min



1

1−pii : pii= 1 

follows easily bycomparing the diagonalentriesinequationM = I−α(I −P ) andusingthatM isdoublystochastic. 2 Welet1= 11T _{bethematrixfullofones.Whenitssizeisimportant,weshallwrite}

1k todenotethek-by-k matrixfullof ones.

Remark2.2. Thecharacterizationoftheidempotentdoublystochasticmatricesisknown (see[3,5]or[8, Theorem I.2.]):IfQ isadoublystochasticidempotent,thenQ is permu-tationallysimilartoablock-diagonalmatrix

(_k1 11k1)  (_k1 21k2)  . . .(_k1 p1kp) forsomepositiveintegerp andsomepositiveintegerski.

Now itis easy to characterize whichpermutation matrices are maximal. Wedenote by Sn the symmetric group of degree n. If σ ∈ Sn we let P (σ) be the corresponding

permutationmatrix.

Theorem2.3. Letσ∈ Sn.Then P (σ) ismaximalin Mn ifandonly if σ isaproductof

disjointtranspositions.

Proof. Assume thatσ∈ Sn is theproduct ofdisjoint transpositions.Then P (σ)2 = I

and so P (σ) = I− 2P where P = 1₂(I − P (σ)) isan idempotent. By Proposition 1.2,

P (σ) ismaximal.

Conversely,assumethatP (σ) isamaximalmatrix.ThenbyTheorem 2.1thereisan idempotentdoublystochasticmatrixP suchthat

P (σ) = I− α(I − P ), α > 0.

Clearly,P (σ) isnotanidempotent,soα= 1.NotethateachrowandcolumnofP (σ) has

onlyoneentrydifferentfromzero.Hence,eachrowandcolumnofP = _α1P (σ)−(1−α)_α I

has at most two nonzero entries and one of them is in the main diagonal. Let i ∈ {1,. . . ,n}.Now, ifP hasjustonenonzeroentry inthei-th row,thenit isinthemain diagonal. Since P is a doubly stochastic matrix this entry is 1. If P has two nonzero entriesintherowi, thenaccordingtoaclassificationofdoublystochastic idempotents

giveninRemark 2.2,thesetwoentriesareequalto 1₂.AtleastonerowofP doescontain

1

2 forotherwise,P = P (σ)= I,a contradiction.Thisimpliesthatα = 2,andhenceσ is

aproductofdisjoint transpositions. 2

Recall thatevery doublystochastic matrix is aconvex combination of permutation matrices. However, not every maximal doubly stochastic matrix can be obtainedas a convex combinationofmaximalpermutationmatrices.

Example 2.4.There exists a maximal doubly stochastic matrix which is not a convex combinationofmaximalpermutationmatrices.Toseethis,considerthedoublystochastic idempotent P =1₃13.ByTheorem 2.1thematrix

M = I−4₃(I− P ) = 1 9 _{1 4 4} 4 1 4 4 4 1 

is maximaldoublystochastic. ByTheorem 2.3 theonlymaximalpermutationmatrices are P (12),P (13),andP (23).So,if

M = α1P (12) + α2P (13) + α3P (23), αi∈ [0, 1],



αi= 1,

then iteasilyfollowsthatα1= 1/9 andα1= 4/9,a contradiction.

Withthenextexamplewe showthatthepolytopeofthedoublystochastic matrices has two adjacentvertices thataremaximal matrices. However,theedge betweenthem onlyhasminimalmatrices.

Example 2.5.There exist two maximaldoubly stochastic matricessuch thattheentire line between them consistsof minimal matrices.Namely, as notedinTheorem 2.3, the matrices P (12) andP (13) are maximal.Butthedoublystochastic matrices

Bt= tP (12) + (1− t)P (13)) = _{0 t (1−t)} t (1−t) 0 (1−t) 0 t , t∈ (0, 1),

areminimal.Namely,thecharacteristicpolynomialofBtequalsp(λ)= (λ− 1)(1− 3t+

3t2− λ2).Thezerosofthesecond factorequal

λ2,3=±  3  t−1 2 2 +1 4

and are clearlydistinct foreachreal t,and alsodistinct from 1 ift∈ (0,1).Hence,for

t∈ (0,1) thematricesBthavethreedistinctrealeigenvalues(oneofthemis1),sothey

3. Maximalcentralizerswithindoublystochastic matrices

Within this sectionwe will investigatethe following morerestrictedproblem: What are the doubly stochastic matrices with maximal centralizer within the set of doubly stochasticmatrices?Wewillrequirethefollowingwell-knownfact(recallthat1 denotes thecolumnvectorfullofones).Wegiveahintoftheproofforthesakeofconvenience.We acknowledgethatthe mainideacame from http :/ /math .stackexchange .com /questions / 70569 /span-of-permutation-matrices.

Lemma 3.1. The algebra generated by doubly stochastic matrices is the same as their linearspanandequalsthespaceofallmatriceswith1 as arightand1T _asaleft

eigen-vector.

Proof. The firstpartfollows sincedoubly-stochastic matricesare aconvex hullof per-mutation matrices (see [1]), which are closed under multiplication. For the last part, a doublystochasticmatrixhas1 asarightand1T _{asalefteigenvector.Samethenholds}

for eachmatrix from Lin_R(Ωn) (here and throughout,given aset Ξ⊆ Mn, we denote

by Lin_R(Ξ) the smallest real vector subspace of Mn which contains Ξ) which bounds

aboveitsdimensiontodim Lin_R(Ωn)≤ 1+ (n− 1)2.Conversely,onecancheck thatthe

1+ (n− 1)2 _{permutationmatrices comingfrom thepermutations id, (1,r),(1,r,s) for}

1= r = s= 1,arelinearlyindependent. 2

Withthehelpofthepreviousresultwecanprovethefollowinglemma,whichwillbe usedseveraltimesinthesequel.

Lemma3.2. Thereexistsorthogonal matrixU ∈ Mn(R) such that

UTΩnU ⊆ 1 ⊕ Mn−1(R).

Moreover,

UT(Lin_RΩn)U =R ⊕ Mn−1(R).

Proof. ThereexistsarealorthogonalmatrixU whosefirstcolumnequals √1

n1.Hence,if

e1,. . . ,enisastandardbasisofcolumnvectorsinRnthenU e1= √1_n1.SinceUT = U−1

weseethatifA isdoublystochastic,thenUT_{AU e1= e}

1andeT1UTAU = eT1,wherefrom UT_{AU∈ 1⊕ M}

n−1(R).

Toprovethe last equalityof theLemma,note thateverymatrixA ∈ R⊕ Mn−1(R)

has (eT

1,e1) as a left/right eigenvector, hence U AUT has (√1_n1T,√1_n1) as a left/right

Corollary 3.3. A doubly stochasticmatrix A∈ Mn, n≥ 2,commutes with every doubly

stochastic matrixifand onlyif

A = d1n+ (1− nd)I, d∈  0, 1 (n−1)  . (3)

Proof. LetU beaunitarysuchthatUTΩnU ⊆ 1⊕ Mn−1(R).IfA∈ Ωn commuteswith

ΩnthenclearlyUTAU commuteswitheverymatrixfromUT(LinRΩn)U =R⊕Mn−1(R),

wherefromUT_{AU = 1⊕ (λI}

n−1)= λIn+ (1− λ)E11forsomescalarλ (here,Eij denotes

the standardmatrix unit).Thus, A= U (λI + (1− λ)E11)UT = λI +(1−λ)_n 1n.This is

entrywisenonnegative ifand onlyifd:= 1−λ_n ∈ [0,_n−11 ]. 2 Here isarestatement ofthepreviousCorollary.

Corollary 3.4. Foradoubly stochastic matrixA one has C(A)∩ Ωn = Ωn if andonly if

A= d1+ (1− nd)I for somed∈0, 1 (n−1)



.

As anotherimmediatecorollary,obtainedafterchoosingd= 1

n inCorollary 3.3:

Corollary 3.5. If a doubly stochastic matrix A∈ Ωn commutes with B ∈ LinRΩn,then

A also commutes with 1

n1− βB for everyreal β.

Thelemmabelowiscrucialtoclassifywhichdoublystochasticmatriceshavemaximal doublystochasticcentralizer.

Lemma 3.6. LetA,B be doubly stochastic.Then C(A)∩ Ωn ⊆ C(B)∩ Ωn if andonly if

C(A)∩ LinR(Ωn)⊆ C(B)∩ LinR(Ωn).

Proof. Assume C(A)∩ Ωn ⊆ C(B)∩ Ωn. Take any X ∈ C(A)∩ LinRΩn. Then, X is

a real matrix so X =ˆ _n11− λX has all entries positive if λ ∈ R is sufficiently small, nonzero. Assuch,X isˆ doubly-stochastic, andhence,byCorollary 3.5,Xˆ ∈ C(A)∩ Ωn.

BytheassumptionsthenalsoXˆ ∈ C(B) wherefrom,againbyCorollary 3.5,X ∈ C(B)∩

Lin_R(Ωn).Theconverseimplication istrivial. 2

WesaythatadoublystochasticmatrixB hasmaximalcentralizerwithinΩn if

CΩn(B) :=C(B) ∩ Ωn= Ωn. Such matriceswereclassifiedinCorollary 3.5.

We say that B has strictly maximal centralizer within Ωn if CΩn(B)  Ωn and for everydoublystochasticX wehavethatCΩn(B) CΩn(X) impliesCΩn(X)= Ωn.Below we classifythose matrices.

Theorem3.7.Adoubly stochasticmatrixA hasstrictlymaximalcentralizerwithin Ωn if

andonly ifA= d1+ (1− nd)I + μQ whereQ∈ Mn(R) isanontrivial idempotentwith

vanishing rowandcolumn sumsandd,μ∈ R arechosensothat A≥ 0,entrywise.

Proof. Again choose a real orthogonal matrix U such that UT_Ω

nU ⊆ 1⊕ Mn−1(R).

Assume first A ∈ Ωn has strictly maximal centralizer within Ωn. Choose any matrix

UTXU ∈ R⊕ Mn−1(R) suchthat

C(UT_{AU )∩ (R ⊕ M}

n−1(R))  C(UTXU )∩ (R ⊕ Mn−1(R)). (4)

BysubtractingasuitablescalarmatrixwecanachievethatUTXU∈ 0⊕Mn−1(R) while

notaffectingtheproperty(4).SinceU ∈ Mn(R) wehavethatX ∈ Mn(R).Hence,there

exists asmall enough positive number λ such that X1 = 1_n1n − λX has nonnegative

entries.Recall thatUT₁₁T_{U = U}T₁

nU = nE11 so UTX1U ∈ 1⊕ Mn−1(R).Therefore, X11= 1 and1TX1 = 1T and henceX1 isdoublystochastic. Clearly, (4)holds for X1

aswell (becauseUT_X

1U = E11− λUTXU )andisequivalentto C(A) ∩ LinR(Ωn) C(X1)∩ LinR(Ωn).

Lemma 3.6 then implies CΩ_n(A)  CΩn(X1) which due to strict maximality of A and

Corollary 3.4 implies that X1 = d1n + (1− nd)I for suitable d ≥ 0. We deduce that

UTAU hasmaximalcentralizerwithinR⊕ Mn−1(R).

Conversely,if UT_{AU ∈ R⊕ M}

n−1(R) hasmaximal centralizer withinR⊕ Mn−1(R)

then, after subtracting form A a suitable scalar,multiplying the resultingmatrix with smallenoughλ> 0 andadding _n11n weobtainadoublystochasticmatrixA1with the

same relative centralizer as A and by Lemma 3.6 A1 has strictly maximal centralizer

withinΩn.

Therefore, it suffices to classify A ∈ Ωn such that C(UTAU )∩ (R⊕ Mn−1(R)) is

maximal within R⊕ Mn−1(R) but not equal to R⊕ Mn−1(R). With the help of Proposition 1.2itistheneasytoseethatthesolutionisUTAU = 1⊕ (μP1+ νIn−1)=

νI + (1− ν)E11+ μP for some nontrivial idempotent P = 0⊕ P1 ∈ 0⊕ Mn−1(R)

and suitably chosen real scalars μ,ν. Since P e1 = 0 and eT

1P = 0 we get that A = d1+ (1− nd)I + μQ where d = 1−ν_n and where Q= U P UT _{∈ M}

n(R) is anontrivial

idempotentwithQ1= 0 and1T_{Q= 0 (i.e., withvanishingrowandcolumnsums). 2}

Remark 3.8.The characterization of minimal doubly stochastic matrices is more in-volved,sincetheset ofminimaldoublystochasticmatricesisopenwithinΩn.

To see this it suffices to show that the set of all nonderogatory matrices of Mn is

openinMn,orequivalentlythatthesetofderogatorymatricesisclosedinMn.Assume

otherwise.Then,therewouldexistanonderogatorymatrixA whichwouldbealimitofa sequenceofderogatorymatricesAm.Passingtoasubsequencewecouldachievethatall

theirminimalpolynomialswouldhaveafixeddegreek < n.Hence,minimalpolynomials ofaAmequal

fm(x) = (x− λ1m)i1m. . . (x− λtm)it,m

for someindex t= t(m)∈ {1,. . . ,k} andexponentsi1,m,. . . ,itm with sumequalto k.

Since |λim|≤ Am m→∞

−−−−→ A weseethatthecoefficientsofminimalpolynomialsare bounded. Passing again to asubsequence we canachieve that all coefficientsof monic polynomials fm(x) converge sothatminimal polynomials of Am alsoconverge to some

polynomial f (x) of degree k. By continuity, f (A) = lim fm(Am) = 0, hence A is not

nonderogatory,a contradiction.

So, characterization of minimal doubly stochastic matrices would have to include, for example thematrices of the form given in(5) below.Let n= 3 and P :=

_{0 0 1}

1 0 0 0 1 0

. ObservethatP iscyclic,hencenonderogatoryandsominimal. Sincethesetofminimal matrices withinΩ3 isopen,wesee that

_{a b 1−a−b}

1−c−d c d

d−a+c 1−b−c a+b−d

(5) is a minimal doublystochastic matrix provided thata,b,c,d aresufficiently close to 0 and eachentry isnonnegative.

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