On bundles of matrix pencils under strict equivalence

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Linear Algebra and its Applications

journal homepage:www.elsevier.com/locate/laa

On bundles of matrix pencils under strict equivalence

Fernando De Terán^∗, FroilánM. Dopico

DepartamentodeMatemáticas,UniversidadCarlosIIIdeMadrid, Avda. Universidad30,28911Leganés,Spain

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

Articlehistory:

Received21April2022 Accepted31October2022 Availableonline4November2022 Submittedby C.Mehl

MSC:

15A22 15A18 15A21 15A54 65F15

Keywords:

Matrix Matrixpencil Matrixpolynomial Spectralinformation Strictequivalence Kroneckercanonicalform Jordancanonicalform Orbit

Bundle Openset

Bundlesofmatrixpencils(understrictequivalence)aresets ofpencilshavingthesameKronecker canonicalform, upto theeigenvalues(namely, theyare aninfiniteunionof orbits under strict equivalence). The notion of bundle for matrix pencilswasintroducedinthe1990’s,followingthesamenotion formatricesundersimilarity,introduced byArnoldin1971, and it has been extensively used since then. Despite the amount of literature devoted to describing the topology of bundles of matrix pencils, some relevant questions remain stillopeninthiscontext.Forexample,thefollowingtwo:(a) provideacharacterizationfortheinclusionrelation between theclosures (inthestandard topology)of bundles;and(b) arethebundlesopenintheirclosure?Themaingoalofthis paperisprovidinganexplicitanswertothesetwoquestions.

Inordertogetthisanswer,wealsoreviewand/orformalize somenotions andresults already existing in theliterature.

Wealso provethat bundlesof matricesunder similarity,as wellas bundlesof matrixpolynomials(definedas thesetof m× n matrixpolynomialsofthesamegradehavingthesame

✩ ThisworkhasbeensupportedbytheAgenciaEstataldeInvestigaciónofSpainthroughgrantsPID2019- 106362GB-I00 MCIN/ AEI/10.13039/501100011033/ and MTM2017-90682-REDT, and by the Madrid Government(Comunidad deMadrid-Spain)undertheMultiannualAgreementwithUC3Minthelineof ExcellenceofUniversityProfessors(EPUC3M23),andinthecontextoftheVPRICIT(RegionalProgramme ofResearchandTechnologicalInnovation).

* Correspondingauthor.

E-mailaddresses:[email protected](F. De Terán),[email protected](F.M. Dopico).

https://doi.org/10.1016/j.laa.2022.10.029

0024-3795/©2022TheAuthor(s). PublishedbyElsevierInc. ThisisanopenaccessarticleundertheCC BY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Closure Majorization

spectralinformation,uptotheeigenvalues)areopenintheir closure.

©2022TheAuthor(s).PublishedbyElsevierInc.Thisisan openaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Orbits of matrices and matrix pencils arise as a natural object when dealing with equivalencerelationsandtheircanonicalforms.Orbitsofmatricesundersimilaritywere introducedbyArnoldin[1],andinthiscasethecanonicalformisthewell-knownJordan Canonical Form (JCF). More precisely, the orbit (under similarity)of agiven matrix consists of all matriceswith thesame JCF.Inother words, it isthe orbitof thegiven matrix (withsize,say,n× n)undertheaction ofsimilarityof thegenerallineargroup GL_n(C),consistingofalln× n invertiblematriceswithcomplexentries,onthesetofall n× n matriceswithcomplexentries, C^n×n,namely

GL_n(C)×^Cn×n → ^Cn×n (P, A) → P AP⁻¹.

In the case of general (unstructured) matrix pencils (namely, pairs of matrices of size m× n),similarityisreplacedbytheso-calledstrictequivalence,thatis

GLm(C)× GLn(C)×^Cm×n×^Cm×n → ^Cm×n×^Cm×n (P, Q, A, B) → (P AQ, P BQ),

andthecanonicalformunderthisrelationistheKroneckerCanonicalForm (KCF)(see [21,Ch. XII,§5]).Thus theorbitofagivenpencilconsistsof allpencilswith thesame KCF.

The notion of orbithas some limitations when studying the change of the spectral information, becausetheeigenvaluesof the matrixor thematrix pencilmust be fixed.

Butif,forinstance,weareinterestedinanalyzingthechangeofthespectralinformation undersmall changesintheentries ofthematricesA and B above,we shouldallowthe eigenvaluesto change,sincethisiswhathappensundergenericperturbations.Inorder toovercomethislimitation,Arnoldintroducedthenotionofbundle inhis1971paper[1].

Abundleistheunion(usuallyinfinite)ofallorbitsthathavethesameJCF(orthesame KCFforpencils)uptothespecificvaluesoftheeigenvalues.Moreprecisely,andtaking thewords byArnold forthecaseofmatrices,“abundleistheset ofallmatriceswhose Jordan normal formsdiffer onlybytheir eigenvalues,butforwhich thesets ofdistinct eigenvaluesandtheordersoftheJordanblocksarethesame”.Forinstance,forthematrix

A =

⎡

⎢⎣

λ0 1 0 0 λ0 0 0 0 μ0

⎤

⎥⎦ ,

consistingofoneJordanblockwithsize2 associatedwiththeeigenvalueλ0andoneJor- danblockwithsize1 associatedwiththeeigenvalueμ0,withλ0= μ0,thecorresponding bundleistheset

B(A) :=

⎧⎪

⎨

⎪⎩P

⎡

⎢⎣

λ 1 0

0 λ 0

0 0 μ

⎤

⎥⎦ P⁻¹ : λ, μ∈^C, λ= μ, P ∈ GL3(C)

⎫⎪

⎬

⎪⎭.

Thenotionofbundlecanbeeasilyextendedtomatrixpencils,namelyabundleistheset ofmatrixpencilswith thesameKCF,upto theeigenvalues.Bundles ofmatrixpencils havebeenconsideredinmanypapers,mostlyinthelast30 years,like[7,15,17–20,29,31].

Someofthese referencesdealwith bundlesofstructured pencils,namelythoseenjoying some particular symmetry in the coefficient matrices, A and B, of the pencil (A,B) (like alternating, (skew-) Hermitian, (anti-)palindromic, or (skew-)symmetric, see, for instance, [26] for the definition of all these structures), and in these cases the strict equivalence relationis replaced bythecongruenceor ∗-congruence relation. Evenbun- dles of unstructured matrices underthe relation of congruence(or ∗-congruence)have beenconsideredintheliterature,mostlyarisingfrom structuredmatrixpencilslike,for instance,in[9,13,14]. However,structuredpencilsareoutoftheanalysis carriedoutin thepresentwork,andweconsider onlygeneral(unstructured)matrixpencils.

Thecanonicalformsmentionedabove(namely,theJCF formatricesortheKCFfor matrixpencils) are therepresentatives of theequivalence classes ofmatrices or matrix pencilsundertheactionofsimilarityor strictequivalence(or congruence/∗-congruence in the structured case), respectively, (see, for instance, [18] for more information on these canonicalformsintheunstructured case,and[32] for structuredmatrixpencils).

Arelevantquestion,notonlytheoretically,butalsoforappliedandnumericalpurposes (for instance, in the computation of the canonical forms) is to determine which are themost“likely”representatives ofthesecanonicalforms.Or,moreprecisely,giventwo differentrepresentativesofthecanonicalform,todeterminewhetheroneofthemismore

“likely”thantheother ornot.Inorder toanswerthisquestion,thestandardapproach, thathasbeenfollowed,forinstance,in[7,8,10–12,14,15,17–19,29] istotranslateittothe context of orbits or,more ingeneral, of bundles. To be moreprecise, and focusing on strict equivalence of matrix pencils, each bundle is associated with a particular pencil L(λ) or,moreingeneral,withtheKCFofL(λ),whichisdetermineduptothevaluesof theeigenvalues.Then,giventwobundlesB(L1) andB(L2),wesaythattheKCFofL1(λ) is“morelikely”(or“moregeneric”)thantheoneofL₂(λ) iftheclosureofthebundleof L2(λ) isincluded intheclosure ofthe bundleof L1(λ) (namelyB(L2)⊆ B(L1), inthe notationthatisused throughout themanuscript),where theclosuresare consideredin thestandard topologyof C^m×n×^Cm×n, which isidentified with C^2mn. Therefore, the inclusion relations of bundle closures determine the “likelihood” of the corresponding canonicalforms.

Fortherelationofsimilarityofmatrices(namely,thelikelihoodoftheJCF),acharac- terizationoftheinclusionrelationbetweenorbitclosuresisknownsincethe1980’s[6,28].

Acharacterizationfortheinclusionrelationbetweenorbitclosuresofmatrixpencilsun- der strict equivalence wasobtained in[2] and[30], and for bundle closures ofmatrices undersimilarity,acharacterizationfortheinclusionrelationispresentedin[18,Th.2.6].

However, we havebeen unable to find intheliterature any explicitcharacterization of theinclusionrelationbetweenbundleclosuresofmatrixpencilsunderstrictequivalence, even though Theorem 3.3 in[18] providesan explicit characterization of the so-called

“covering” relation between bundle closures of matrix pencils (where covering means that B(L2)⊆ B(L1) andthere is noany L3 suchthatB(L2)⊂ B(L3)⊂ B(L1), and ⊂ meansstrict inclusion).

In theargumentsdescribed at theend ofthe last-but-oneparagraph, itis implicitly assumed thattheKCFofL(λ) isthe“generic”oneinB(L).Letusrecallthatasubset, S0,ofagivensetS (inatopologicalspace)iscalledgenericinS ifS0isopenanddensein S. Therefore,theassumptiononthegenericity oftheKCFjustmentionedisequivalent to saythat B(L) isopen and dense inits closure. Clearly, B(L) isdense inB(L). The propertyofbeingopeninitsclosureiswell-knownfororbitsofvarietiesundertheaction ofagroup(see,forinstance,[23,p.60]).Thisisthecase,forinstance,ofmatricesunder similarityandmatrixpencilsunderstrictequivalenceandcongruence.However,wehave not foundintheliterature acorresponding resultforbundles so,upto ourknowledge, thequestiononwhetherthebundlesareopen intheirclosureisstillopen.

Summarizing, thefollowing tworelevantquestionsarise:

Q1. ToprovideacharacterizationfortherelationB(L2)⊆ B(L1) tohold,fortwogiven matrixpencilsL1 andL2.

Q2. Toprovethatbundles areopenintheirclosure.

Themain goalofthepresentworkistoprovide ananswerto theprevioustwo ques- tions.Togetthisanswer,werevisitsomenotionsthatarealreadyintheliterature,like the notionof “coalescence”of eigenvalues,as well as some results,like Theorem 7.5 in [29],whichisproventobe false.

Asanasideresult,wearealsoabletoprovethatbundlesofmatricesundersimilarity, aswellas bundlesofmatrixpolynomials,definedassetsofmatrixpolynomialswiththe samesizeandgrade(seeSection4.1forthisnotion)havingthesamespectralinformation, uptothespecificvaluesoftheireigenvalue(seeSection4.2),arealsoopenintheirclosure.

Therest ofthepaper isorganizedasfollows. InSection2weintroducethenotation andbasicnotionsusedthroughoutthepaper,togetherwithsomealreadyknownresults thatare used later.Section3is devotedto thesolutionof questionQ1 above,whereas in Section 4 we provide an affirmative answer to question Q2 for bundles of matrix pencilsunderstrictequivalence,aswellasformatricesundersimilarity(inSection4.1), and matrix polynomials of higherdegree(in Section4.2).Finally, Section5presents a

summary of the main contributions of the paper, together with some lines of further relatedresearch.

2. Basicdefinitionsandnotation

Weusethe followingnotation throughout thepaper. By C^m×n we denote theset of m× n matriceswithcomplexentries,whereasGLn(C) denotesthesetofn× n invertible matriceswith complexentries.Also, C=C ∪ {∞}.ByIk we denote thek× k identity matrix,andν(A) denotesthedimensionofthe(right)nullspaceofthematrixA.Instead of using the representation of matrixpencils as pairsof matrices of the samesize, we willrepresentamatrixpencilasL(λ)= λB + A,withA,B∈^Cm×n,namelyasamatrix polynomialof degree1 inthevariable λ. Weoften denote matrixpencilswith asingle capitalletter(usuallyL andM )and,forthesakeofsimplicity,inthenotationfornotions associatedwithamatrixpencilweomitthevariableλ,andwritejustL insteadofL(λ).

Therank ofthepencilL(λ) (thatissometimesfoundintheliteratureunderthename

“normalrank”),denotedbyrank L, istherankof L(λ) consideredasamatrixoverthe fieldof rationalfunctionsin thevariable λ. Inother words, itis thesize of the largest non-identicallyzerominor ofL(λ).

Block-partitionedmatrices(or matrixpencils) willappearfrequentlythroughoutthe manuscript,andtheblocksindicatedwith∗ arenotrelevantinthearguments,develop- ments,orresults.Forablockdiagonalpencil(ormatrix)withdiagonalblocksA₁,. . . ,A_k weuseeitherthenotationdiag(A₁,. . . ,A_k) ork

i=1A_i. 2.1. The KFC,orbitsand bundles

LetusrecallthattheKCFofamatrixpencilL(λ) (thatwedenote byKCF(L))isa blockdiagonalpencil,whose diagonal (“canonical”)blockscanbe of thefollowing four forms(see,forinstance,[21,Ch. XII,§5]):

• Jordan blocksassociatedwith afiniteeigenvalueμ, namelyλIk+ Jk(μ),for k 1, where

Jk(μ) :=

⎡

⎢⎢

⎣

−μ 1 . .. ...

−μ 1

−μ

⎤

⎥⎥

⎦

k×k

.

• Jordan blocksassociated withthe infiniteeigenvalue, namely λN_k + I_k, fork  1, where:

Nk:=

⎡

⎢⎢

⎣

0 1

. .. ...

0 1

0

⎤

⎥⎥

⎦

k×k

.

• Rightsingularblocks,fork 0:

R_k(λ) =:

⎡

⎢⎢

⎣ λ 1

λ 1

. .. ...

λ 1

⎤

⎥⎥

⎦

k×(k+1)

.

• Leftsingularblocks,R_k(λ)^,fork 0.

TheKCFofL(λ) isdetermineduptopermutationofthediagonalblocks.

Let us note that R₀(λ) is a null column, whereas R₀(λ)^ is a null row. The KCF reveals all theinvariants ofa matrixpencilunderstrict equivalence, namely theset of distinct finite and infinite eigenvalues together with the numberand thesizes of their associated Jordan blocks, and the number and the sizes of the right and left singular blocks.

Then, μ∈^C is afinite eigenvalue of L(λ) if KCF(L)contains, at least, oneJordan block associated with μ, and L(λ) has the infinite eigenvalue if KCF(L) contains, at least,oneJordanblockassociatedwiththeinfiniteeigenvalue.ThepencilL(λ) issaidto beregular ifthereareneitherrightnorleftsingularblocksinKCF(L)(thisisequivalent to say thatL(λ) issquare and det L(λ) is anon-identicallyzeropolynomial).ByΛ(L) we denote the spectrum of the pencil L(λ) (namely, the set of distinct eigenvalues of L(λ),both finiteandinfinite).

We denote by W (μ,L) = (W1(μ,L),W2(μ,L),. . .) the Weyr characteristic of the eigenvalueμ inthem× n matrixpencilL(λ).Inotherwords,whenμ∈^C(respectively, μ=∞),Wi(μ,L), fori 1,is thenumberofJordanblocksλIk+ Jk(μ) (resp.,λNk+ Ik), with k  i, in KCF(L), (see, for instance,[4]). If μ is not an eigenvalueof L(λ), then W (μ,L)= (0,0,. . .). Also,r(L)= (r0(L),r1(L),. . .) and(L)= (0(L),1(L),. . .) denote,respectively, theWeyrcharacteristic ofthe rightand left singularstructure.In otherwords,r_i(L) (respectively,_i(L))isthenumberofright(resp.,left)singularblocks Rk(λ) of size k× (k + 1) (resp.,Rk(λ)^ of size (k + 1)× k), with k  i, inKCF(L).

In particular, r0(L) = n− rank L and 0(L) = m− rank L. Note that W (μ,L),r(L), and(L) arelistsofnon-increasingintegers.WeusethenotationL1≺ L2 todenotethe majorization of two lists of non-increasing integers,namely: j

i=1L1(i)j

i=1L2(i), forallj 1,assumingthatthelistsstartwith i= 1.

For μ ∈ ^C and ε > 0, the ε-neighborhood of μ is defined as B(μ,ε) := {z ∈ ^C :

|z − μ|< ε},whereasifμ=∞,weset B(∞,ε):={z ∈^C : |z|> ε⁻¹}).

Following the notation in [29], Φ denotes the set of all one-to-one mappings of C to itself. We also use the notation Ψ for the set of all mappingsfrom C to itself (not necessarilyone-to-one).Then,ifKL(λ):= KCF(L) (wherethecanonicalblocksaregiven inany order), andψ∈ Ψ,wedenote byψ(L) anypencilwhichis strictly equivalentto the pencil obtained from K_L(λ) after replacing the Jordan blocks associated with the eigenvalueμ∈^CbyJordanblocks ofthesamesizeassociatedwiththeeigenvalueψ(μ),

forany eigenvalueμ ofKL(λ).Note thatthis pencil ψ(L) isnot uniquelydetermined, butallpencilsψ(L) arestrictlyequivalent toeachother.

ForagivenmatrixpencilL(λ)= λB + A,withA,B ∈^Cm×n,weset:

O(L) := {λP BQ + P AQ : P ∈ GLm(C), Q∈ GLn(C)} (orbit of L(λ)),

B(L) := 

ϕ∈Φ

O(ϕ(L)) (bundle of L(λ)).

Note that O(ϕ(L)) is well defined, regardless of the particular pencil ϕ(L), since, as mentionedabove,allpencilsϕ(L) arestrictlyequivalent.

Bydefinition, bundles are theunion of orbitsof all matrixpencils having thesame KCF up to the eigenvalues. Note that this union is infinite provided that the pencils have, at least, one eigenvalue. However, if KCF(L) contains only blocks of the form R_k(λ) and/orR_k(λ)^,thenB(L)=O(L).Somespecialattentionshouldbepaidtothe infiniteeigenvalue.More precisely,the reasonforconsideringmaps over C is,precisely, toincludetheinfiniteeigenvalueinthebundles.Forinstance,ifL(λ) isoftheform

L(λ) = λ

⎡

⎢⎣

1 0 0 0 1 0 0 0 1

⎤

⎥⎦ +

⎡

⎢⎣

−μ 1 0 0 −μ 0 0 0 −μ

⎤

⎥⎦ , μ= μ

(namely, thepencil has two different finite eigenvalues μ and μ with Jordan blocks of sizes2 and1,respectively)then

B(L) =



λP Q + P

_{−a 1}

0 −a

−a



Q : a,a ∈^C, a= a, P, Q ∈GL3(C)



 λP

_{1 0}

0 1 0

 Q + P

_{−a 1}

0 −a 1



Q : a∈^C, P, Q∈GL3(C)



 λP

_{0 1}

0 0 1

 Q + P

_{1 0}

0 1

−a



Q : a∈^C, P, Q∈GL3(C)

 .

The first set in the union above corresponds to a direct sum of two Jordan blocks associated with a couple of finite (different) eigenvalues, the second one corresponds to aJordan block of size 2 associatedwith afinite eigenvalue,together with aJordan blockofsize 1 associated with theinfiniteeigenvalue, and thethirdset corresponds to aJordanblockof size2 associated withtheinfinite eigenvalue,togetherwith aJordan blockof size1 associatedwithafinite eigenvalue.

WewillusethestandardnotationS fortheclosure,inthestandardtopology,ofthe setS.Inthiscontext,thesetofmatrixpencilsλB + A,withA,B∈^Cm×n,isidentified with C^2mn,andweconsiderthestandardtopologyinthisset.Fortheclosureoftheorbit andthebundleofamatrixpencilL(λ) weusethenotationO(L) andB(L),respectively.

2.2. Coalescenceofeigenvalues

The notionofcoalescence of eigenvalues,whichis keyto describethe inclusionrela- tionshipsforclosuresofbundlesofmatricesandmatrixpencils,hasbeenusedinprevious references,including[17,18].Westatehereaformaldefinition,whichisequivalenttothe onementionedin[18,Th.3.3-(5)].Werecallthattheunionoftwolistsofnon-increasing integers(liketheWeyrcharacteristics),sayL1 andL2,thatwedenotebyL1∪ L2,con- sistsofanewlistofnon-increasing integerswhichisobtainedbyarrangingallnumbers inL1andL2inanon-increasingorder.Forinstance,ifL1= (5,2) andL2= (6,3,3,2,1), then L1∪ L2= (6,5,3,3,2,2,1).

To understandthefollowing definition,we recallthatΨ denotestheset ofmappings from Ctoitself.

Definition 1.(Coalescence of eigenvalues). Let L(λ) be a matrix pencil with distinct eigenvaluesμ1,. . . ,μs,andletψ∈ Ψ.Then,ψc(L) isanymatrixpencilofthesamesize as L(λ) satisfyingthefollowing threeproperties:

• r(ψ_c(L))= r(L),

• (ψ_c(L))= (L),and

• W (μ,ψ_c(L))=

μi∈ψ⁻¹(μ)W (μ_i,L),forallμ∈^C.

We say that the eigenvaluesμi1,. . . ,μid of L(λ) have coalesced to the eigenvalue μ in ψc(L) ifψ⁻¹(μ)={μi1,. . . ,μid}∪ S,withS∩ Λ(L)=∅.

Remark2.Thematrixpencilψc(L) inDefinition1isnotuniquelydefined,butallpencils ψc(L),forsomegivenL(λ) andψ,arestrictlyequivalenttoeachother,sincetheyallhave thesameKCF(asithappenswithψ(L)).Moreover,notethatB(ψc(L))=B( ψc(L)),for any ψ,ψ∈ Ψ suchthatψ(μ)= ψ(μ) ifandonlyifψ(μ) = ψ(μ),forany μ= μ.

Coalescenceofeigenvaluesisawayof“gathering”eigenvaluesbytakingtheunionof theirWeyrcharacteristics.Thefollowing exampleaimsto illustratethisnotion.

Example 3.LetL(λ) bethefollowingpencil,alreadygiveninKCF:

L(λ) = diag (R3(λ), R1(λ), λI2+ J2(0), λI2+ J2(0), λI1+ J1(0), λI3+ J3(1), λI2+ J2(1), λI4+ J4(2), R2(λ)^),

so that r(L) = (2,2,1,1), (L) = (1,1,1), and W (0,L) = (3,2), W (1,L) = (2,2,1), W (2,L)= (1,1,1,1).

Letψ :C→^Cbe suchthatψ(0)= ψ(1)= ψ(2)= 1.Thenanypencilψ_c(L) isof the form:

ψc(L) = P· diag(R3(λ), R1(λ), λI9+ J9(1), λI4+ J4(1), λI1+ J1(1), R2(λ)^)· Q,

forsomeinvertiblematricesP,Q.Note,indeed,that,foranyψc(L) asabove,r(ψc(L))= (2,2,1,1)= r(L),(ψc(L))= (1,1,1)= (L),andW (1,ψc(L))= (3,2,2,2,1,1,1,1,1)= W (0,L)∪ W (1,L)∪ W (2,L).

However,ifψ is suchthatψ(0)= ψ(2) = 1,ψ(1) = 5, then anypencil ψc(L) is now oftheform

ψ_c(L) = P · diag(R3(λ), R₁(λ), λI₆+ J₆(1), λI₂+ J₂(1), λI₁+ J₁(1), λI3+ J3(5), λI2+ J2(5), R2(λ)^)· Q,

forsomeinvertiblematricesP,Q.Note,indeed,that,foranyψc(L) asabove,r(ψc(L))= (2,2,1,1) = r(L), (ψc(L)) = (1,1,1) = (L), and W (1,ψc(L)) = (3,2,1,1,1,1) = W (0,L)∪ W (2,L),W (5,ψc(L))= (2,2,1)= W (1,L).

2.3. Some basic results

We are interested in the majorization of the Weyr characteristics of an eigenvalue intwo givenmatrix pencils.This majorizationis definedbyinequalities onthesum of the first elements of the corresponding Weyrcharacteristic, so aformula for this sum wouldbequiteusefultothisend.Suchaformulawillcomefromtherankofcertainbig block-partitioned matrices. For describing this, we follow some of thedevelopments in thePhD.thesis[3], thatweincludehereforthesakeofcompleteness.

GiventhematrixpencilL(λ)= λB + A, withA,B ∈^Cm×n andμ∈^C,wedefinethe followingblock-partitionedmatriceswith d blockcolumnsand d blockrows:

P_μ^d(L) :=

⎡

⎢⎢

⎢⎢

⎢⎢

⎣

L(μ) 0 . . . . . . 0

B L(μ) . .. ...

0 B . .. . .. ... ... . .. . .. L(μ) 0

0 . . . 0 B L(μ)

⎤

⎥⎥

⎥⎥

⎥⎥

⎦

dm×dn

, for d 1. (1)

Notethat

P_μ^d(L) =

 0 0

I_d−1 0



⊗ B + Id⊗ L(μ),

where⊗ denotestheKroneckerproduct.

Ford₁,. . . ,d_s 1 andλ₁,. . . ,λ_s∈^C(differenttoeachother),wedefinethefollowing block-partitionedmatrices:

P_λ^d₁¹_,...,λ^,...,d_s^s(L) :=

⎡

⎢⎢

⎢⎣ P_λ^d1

1(L) 0 0 . . . 0

Q_d2,d1(B) P_λ^d2

2(L) 0 . . . 0

... . .. . .. ...

0 0 . . . Qds,d_s−1(B) P_λ^d_s^s(L)

⎤

⎥⎥

⎥⎦,

where,for2 i s,

Q_di,di−1(B) :=

⎡

⎢⎢

⎣

0 . . . 0 B 0 . . . 0 0 ... . .. ... ...

0 . . . 0 0

⎤

⎥⎥

⎦ ∈^{Cdim×di−1n}.

Lemma4.LetL(λ)= λB + A.Then, forany distinctλ₁,. . . ,λ_s∈^Candd₁,. . . ,d_s 1, ν(diag(P_λ^d₁¹(L), . . . , P_λ^d_s^s(L))) = ν(P_λ^d₁¹_,...,λ^,...,ds_s(L)),

where ν(Z) denotes thedimensionoftheright nullspace ofthematrixZ.

Proof. The result is a consequence of the fact that P_λ^d₁¹_,...,λ^,...,d_s^s(L) is equivalent, by elementary block-row and block-column operations, to the block diagonal matrix diag(P_λ^d₁¹(L),. . . ,P_λ^d_s^s(L)) (because these operations preserve the rank and, as a con- sequence, the dimension of the nullspace, of the matrix). To see this equivalence, let us firstconsider justtwobig blocks,namelys= 2,so thematrixreads P_α,β^d1,d2(L),with α,β∈^Cand α= β. Notethat,foranys∈^C,

sB = s

 1

α− βL(α)− 1 α− βL(β)



. (2)

If E_i,j(σ) denotesthe elementarymatrixoftheappropriatesize having1’sinthemain diagonal, σ inthe(i,j) entryandzeroeselsewhere,thenthematrix



E_d1+1,d1

 1

β− α



⊗ Im



P_α,β^d1,d2(L)



E_d1+1,d1

 1

α− β



⊗ In



isobtainedfromP_α,β^d1,d2(L) addingtoitsd1thblock-columnthe(d1+ 1)stonemultiplied by1/(α− β),andtothe(d1+ 1)stblock-rowthed1thonemultipliedby−1/(α − β).By (2),this matrixhasa0 block inthe(d1+ 1,d1) block position,insteadof theblockB thatappearsinP_α,β^d1,d2(L),buttheblocksinthepositions(d₁+ 1,d₁− 1) and(d₁+ 2,d₁) are oftheform s₁B ands₂B, forsomes₁,s₂∈^C.These blocksareinablockdiagonal that is below the one containing B in the original matrix P_α,β^d1,d2(L), so the previous blockrowand blockcolumnoperations havetakentheblockB tosomemultiplesofB in the lower block diagonal. Because of (2), we canfind appropriate elementary block rowandblockcolumnoperationsthatturntheseblocksinto0,butcreatesomenonzero blocks, which are, again, multiples of B, in the nextlower blockdiagonal. Proceeding recursivelyinthisway,weendupwithamatrixwhichisequaltodiag(P_α^d1(L),P_β^d2(L)) except for a nonzero blockof the form sB, for some s ∈ ^C, in the (d₁+ d₂,1) block- position. Using again (2),an appropriatelinearcombination of thefirst blockrowand thelast(i.e.,the(d1+ d2)thone)blockcolumnwillshrinkthisblockto0,soweendup with diag(P_α^d1(L),P_β^d2(L)),aswanted.

Ifthere aremorethan two blocks, wepartition thematrix P_λ^d₁¹_,...,λ^,...,ds_s(L) intoa 2× 2 blockmatrix,as follows:

P_λ^d₁¹_,...,λ^,...,ds_s(L) =

P_λ^{d1,...,ds−1}

1,...,λ_s−1(L) 0 Q_d

s, d_s−1(B) P_λ^d_s^s(L)



, (3)

with d_s−1 := d₁+ d₂+· · · + ds−1. Using appropriateelementaryblock-row and block- columnoperationsas explainedbeforeforthecases= 2,thismatrixisequivalent to



P_λ^d₁¹_,...,λ^,...,ds−1_s−1(L) 0 0 P_λ^ds

s(L)



. (4)

Tobemoreprecise,inthiscase,insteadof(2),weneedto usetheidentity

sB = s

 1

λ_i− λs

L(λ_i)− 1 λ_i− λs

L(λ_s)

 ,

fori= 1,. . . ,s− 1,and theblock-row andblock-columnoperationsproduce somemul- tiplesof B in the(2,1) block of (3), that,following thesameprocedure as before, will move from one block diagonal to the “lower” one. Then, after a finite number of el- ementary block-row and block-column operations we arrive at (4). From (4) we can proceedinthesamewaywiththeupperleftblockP_λ^d₁¹_,...,λ^,...,d_s−1^s−1(L),andsoon,untilweget diag(P_λ^d₁¹(L),. . . ,P_λ^d_s^s(L)). 

Thefollowing resultprovides aformulaforthe sumofthe firstd termsintheWeyr characteristicof aneigenvalueof amatrix pencil,intermsofthedimension ofthenull spaceofthematrixin(1).

Lemma5.LetL(λ) bean m× n matrix pencilandμ∈^C.Then, foralld 1:

ν(P_μ^d(L)) =

d i=1

Wi(μ, L) + d r0(L),

wherer0(L)= n− rank L.

Proof. It is straightforward to see that, if L(λ) is a strictly equivalent pencil to L(λ), then ν(P_μ^d(L)) = ν(P_μ^d(L)). Therefore, we may assume that L(λ) is given in KCF. Letus decomposeL(λ)= diag(Lr(λ),Lreg(λ),L(λ)), where Lreg(λ) contains all Jordan blocks of L(λ), L_r(λ) contains all blocks of the form R_k(λ), whereas L_(λ) contains the blocks of the form R_k(λ)^. By means of column and row permuta- tions, P_μ^d(diag(L_r,L_reg,L_)) is strictly equivalent to diag(P_μ^d(L_r),P_μ^d(L_reg),P_μ^d(L_)), so ν(P_μ^d(L)) = ν(P_μ^d(diag(Lr,Lreg,L))) = ν(diag(P_μ^d(Lr),P_μ^d(Lreg),P_μ^d(L))) = ν(P_μ^d(Lr))+ ν(P_μ^d(Lreg))+ ν(P_μ^d(L)).

Now, sinceP_μ^d(L) hasfullcolumnrank,for everyμ∈^C,then ν(P_μ^d(L))= 0.Also, sincer₀(L) isthenumberofblocksR_k(λ) inL_r(λ),itisalsostraightforwardtoseethat ν(P_μ^d(Lr))= dr0(L).Finally, ν(P_μ^d(Lreg))=d

i=1Wi(μ,Lreg)=d

i=1Wi(μ,L), where thefirstidentity isalsostraightforwardtoget(seealso[22,p.36]). _

Inthecasewherer0= 0,Lemma5isaconsequenceofthedevelopmentscarriedout in[24,§5].

The Weyrcharacteristic ofthe eigenvalueμ inthe pencilL(λ) is theconjugate par- tition of theSegre characteristic, denotedby S(μ,L) = (S1(μ,L),S2(μ,L),. . .), where S_i(μ,L) is the size of the ith largest Jordan block associated with theeigenvalue μ in KCF(L) (see,forinstance,[4]).Ingeneral, theconjugateofapartition,a,is theparti- tion, denotedbya^, whoseithelement,fori 1,isequalto thenumberof elementsin a whichare greaterthanorequaltoi. Also,thesumofafinite numberofpartitionsis the partitionwhose ithelement isthe sumoftheithelements inallpartitions(adding zeroesattheendofthepartitionsifnecessary).Thefollowingresultcanbefoundin[25, p.6].

Lemma 6.Leta1,. . . ,ak be partitions.Then

 _k



i=1

ai

^

=

k i=1

a_i^, and

 _k

i=1

ai

^

=

k i=1

a^_i.

Thecharacterizationfortheinclusionbetweenorbitclosuresofmatrixpencils,which wasobtainedindependentlyin[2] and[30],and reformulatedin[4],willbe usedseveral times inthepaper. Weincludeithereforcompleteness(thestatementwepresent isin betweentheones of[4, Lemma1.3] and[18,Th.3.1]).

Theorem7.(Characterizationoftheinclusionbetweenorbitclosures).Thematrixpencil P₂(λ) belongs to theclosure of the orbit of the pencil P₁(λ) (in other words, O(P2) ⊆ O(P1))if andonly ifthefollowingrelationshold:

(i) r(P₂)≺ r(P1)+ (h,h,. . .), (ii) (P2)≺ (P1)+ (h,h,. . .),

(iii) W (μ,P1)≺ W (μ,P2)+ (h,h,. . .),forallμ∈^C, where h:= rank P₁− rank P2.

One of the main goals of this paper is to provide a characterizationlike the onein Theorem7forclosuresofbundlesinsteadoforbits.ThiswillbeprovidedinTheorem12.

3. Acharacterizationforthe inclusionofbundleclosuresofmatrixpencils

The main result of this section is Theorem 12, which provides a characterization for theinclusionof bundle closures of matrixpencils. To proveit weuse the following technicalresult.Itisaconsequenceof[3,Teorema2.3],adaptedtotheconditionsinthe statement.Intheproofweusethenotionofreversal ofthematrixpencilL(λ)= λB +A, definedasrevL:= λA+ B.

Theorem8.Let{Mk(λ)}k∈N be asequenceof m× n complexmatrixpencilssuch that:

(i) Mk(λ)∈ B(L),forsome m× n complexmatrix pencilL(λ) andforallk∈^N, (ii) λ_1,k,. . . ,λ_s,k∈^CaredistincteigenvaluesofM_k(λ),fork∈^N,andW (λ_i,k1,M_k1)=

W (λ_i,k2,M_k2), foralli= 1,. . . ,s,andallk₁,k₂∈^N, (iii) {Mk(λ)}k∈N convergestoM (λ), and

(iv) the sequence{λi,k}k∈N convergestoμ,foralli= 1,. . . ,s,where μ∈^C. Thens

i=1W (λ_i,k,M_k)≺ W (μ,M )+ (h,h,. . .),where h:= rank L− rank M.

Proof. Let us first assume that μ ∈ ^C (namely, μ = ∞). Then, for k large enough, λi,k∈^C,forall1 i s.Therefore,wemayassumethatλi,k∈^C,forall1 i s and allk∈^N.Intherestofthiscase,weessentiallyfollow theproof of[3, Teorema2.3].

Setm= (m1,m2,. . .):=s

i=1W (λi,k,Mk) (notethatthisdoesnotdependonk, by condition(ii)inthestatement).BydefinitionofunionofWeyrcharacteristics,foreach d 1,there ared1,. . . ,ds 0 suchthatd1+· · · + ds= d and

d i=1

mi =

d1

j=1

Wj(λ1,k, Mk) +· · · +

ds

j=1

Wj(λs,k, Mk), (5)

forallk∈^N,where 0

j=1Wj(λi,k,Mk):= 0,fori= 1,. . . ,s.

Sincethe pencilsMk(λ) converge toM (λ) and the valuesλi,k converge to μ,for all 1 i  s, taking into account thatd1+· · · + ds = d,we conclude that the matrices P_λ^d_1,k^1,...,d_,...,λ^s_s,k(Mk) (where,if di = 0 forsome1 i s,theblockrow andblockcolumn correspondingtoP_λ^d_i,kⁱ (Mk) isnotpresent)convergetoP_μ^d(M ).Then,bythelowersemi- continuityoftherank,we get

rank P_μ^d(M ) rank Pλ^d1,k^1,...,d,...,λ^ss,k(Mk), (6) fork largeenough.

From Lemma 5, and taking into account that rank Mk = rank L, for allk ∈ ^N, we get,foreach1 i s andallk∈^N:

di

j=1

Wj(λi,k, Mk) + di(n− rank L) = ν(Pλ^di,kⁱ (Mk)) = ndi− rank (Pλ^di,kⁱ (Mk)).

From thisidentity and(5) weobtain,forallk∈^N,

d i=1

mi+ d(n− rank L) =

s i=1

⎛

⎝^di

j=1

Wj(λi,k, Mk) + di(n− rank L)

⎞

⎠

= nd−

s i=1

rank (P_λ^d_i,kⁱ (Mk))

= nd− rank diag(P_λ^d_1,k¹ (Mk), . . . , P_λ^ds

s,k(Mk))

= ν(diag(P_λ^d_1,k¹ (Mk), . . . , P_λ^d_s,k^s (Mk)))

= ν(P_λ^d_1,k^1,...,d_,...,λ^s_s,k(Mk)),

(7)

where thelastidentity isaconsequenceofLemma4.

Now (7),togetherwith(6) andLemma5implythat

d i=1

m_i+ d(n− rank L) = ν(Pλ^d1,k^1,...,d,...,λ^ss,k(M_k))

 ν(Pμ^d(M )) =

d i=1

Wi(μ, M ) + d(n− rank M),

or,equivalently,

d i=1

mi

d i=1

Wi(μ, M ) + d(rank L− rank M),

as wanted.

Now, letus assumethatμ=∞.It is straightforwardto see(but we referotherwise to [27, Remark 4.3]) that, for agiven pencil L(λ) and μ ∈ ^C, the identity W (μ,L) = W (μ⁻¹,revL) holds.It isalsoimmediateto seethatrank (revL)= rank L. Moreover,if {Mk(λ)}k∈ZisasequenceofpencilsconvergingtoM (λ),then{revMk}k∈Zconvergesto revM . SoletL(λ),M (λ),and M_k(λ),as wellas λ_i,k,beas inthestatement.Then λ⁻¹_i,k are eigenvalues ofrevMk, and theyconverge to the eigenvalue0 of revM . Bythe case justprovedforμ∈^C,weconcludethat

s i=1

W (λ⁻¹_i,k, revM_k)≺ W (0, revM) + (h, h, . . .).

But,sinceW (λ⁻¹_i,k,revMk)= W (λi,k,Mk) andW (0,revM )= W (∞,M ), thisimplies

s i=1

W (λi,k, Mk)≺ W (∞, M) + (h, h, . . .),

as claimed. 

Thefollowingresultprovidesatheoreticalanswertothequestionofdecidingwhether agiven matrixpencilbelongstotheclosureofthebundleofanotherpencilor not.

Theorem 9.Let L(λ) and M (λ) be two complex matrix pencilsof the same size. Then M (λ)∈ B(L) ifandonly ifM (λ)∈ O(ψc(L)), forsomemapψ :C→^C.

Proof. Letusfirstprovethe“if”partofthestatement.AssumethatM (λ)∈ B(L).Then, there is a sequence of pencils{Mk(λ)}k∈N such that Mk(λ) ∈ B(L) and {Mk(λ)}k∈N

convergestoM (λ).SinceM_k(λ)∈ B(L),weconclude:

• r(Mk)= r(L) and(Mk)= (L),forallk∈^N.

• AllMk(λ) havethesamenumberofdistincteigenvalues,says,whichisthenumber ofdistinct eigenvaluesofL(λ).Moreover,ifλ₁,. . . ,λ_s andλ_1,k,. . . ,λ_s,k denote,re- spectively,thedistincteigenvaluesofL(λ) andM_k(λ),thenW (λ_i,k,M_k)= W (λ_i,L), forallk∈^Nand alli= 1,. . . ,s.

If we set h := rank L− rank M and σε(M ) = 

μ∈Λ(M)B(μ,ε) then, for all k large enough,thepencilsM_k(λ) satisfy(see, forinstance,Lemma1.1anditsproofin[4]):

(i) r(M )≺ r(Mk)+ (h,h,. . .)= r(L)+ (h,h,. . .), (ii) (M )≺ (Mk)+ (h,h,. . .)= (L)+ (h,h,. . .),

(iii) W (μ,M_k)≺ W (μ,M )+ (h,h,. . .),forallμ∈ (^C− σε(M ))∪ Λ(M).

Now,weare goingtosee that,bytaking asubsequenceof{Mk(λ)}k∈N ifnecessary, we mayassumethat,foranyi= 1,. . . ,s oneofthefollowingconditionsholds:

(C1) {λi,k} convergestoaneigenvalueofM , or

(C2) thereissomeε> 0 suchthatλ_i,k∈^C− σε(M ),forallk largeenough.

Assumethat,for somei= 1,. . . ,s,condition (C2)does nothold.Then, for anyε > 0 there is aninfinite subsequenceof {λi,k} included inσε(M ). Since the spectrumof M is finite,there isasubsequence, {λi,kj} thatconverges to someeigenvalueof M (λ),so {λi,kj} satisfies(C1).Sinces isafinitenumber,wecankeepgoingwiththisprocedure,by takingafinersubsequenceifnecessary,untilweendupwithasubsequenceof{Mk(λ)}, whoseeigenvalues{λi,k} satisfyeither(C1)or(C2), foralli= 1,. . . ,s.

Now,letusassumethattheeigenvaluesλ_i,kconvergeto somedistinctμ₁,. . . ,μ_d,for alli= 1,. . . ,t and t s,whereμ₁,. . . ,μ_d areeigenvaluesofM (λ).Note,however,that notalleigenvaluesof Mk(λ) mustconvergeto eigenvaluesof M (λ),sincesomeof them cansatisfy condition(C2) above,and thatnotevery eigenvalueof M (λ) is necessarily obtainedinthisway,becausesomeneweigenvaluesnotcomingfromeigenvaluesofM_k(λ) couldarise inthelimitM (λ).Inparticular, we areassuming thatthe eigenvaluesλ_i,k, fori= t+ 1,. . . ,s, satisfy(C2)above. Letus gatheralleigenvaluesλi,k thatconverge

tothesameeigenvalueμj ofM (λ) inthefollowingway:wedecomposethesetofindices {1,. . . ,t} as theunionof d disjointsets ofindices, denotedbyI1,. . . ,Id insuchaway that the eigenvalues λi,k with i ∈ Ij converge to μj, for j = 1,. . . ,d. This means, in particular, thatM (λ) has,atleast,d different eigenvalues,namelyμ1,. . . ,μd.

Let L := ψ_c(L), where ψ : C → ^C is such that ψ(λi) = μj, for i ∈ Ij and j = 1,. . . ,d, ψ(λi) = λi, for i = t+ 1,. . . ,s, and ψ(μ) ∈ {μ/ 1,. . . ,μd,λt+1,. . . ,λs}, for μ∈ {λ/ 1,. . . ,λs}.Then,μ1,. . . ,μd,λt+1,. . . ,λs arethedistincteigenvaluesof L,and:

(iv) r(M_k)= r(L)= r(L), (v) (Mk)= (L)= (L),

(vi) W (μ,L)≺ W (μ,M )+ (h,h,. . .),forallμ∈^C. Claim (vi)isaconsequenceofthefollowingfacts:

• Ifμ isnotaneigenvalueof L(λ),then(0)= W (μ,L)≺ W (μ,M )+ (h,h,. . .), since h 0.

• W (μj,L) =

i∈IjW (λi,L) =

i∈IjW (λi,k,Mk)≺ W (μj,M )+ (h,h,. . .), for j = 1,. . . ,d,wherethefirstidentityisaconsequenceofthedefinitionofψc(L),and the majorizationis aconsequenceofTheorem8.

• For the remaining eigenvalues of L, namely λi with i = t + 1,. . . ,s, we have W (λi,L)= W (λi,L)= W (λi,k,Mk)≺ W (λi,k,M )+ (h,h,. . .)= (0)+ (h,h,. . .)≺ W (λi,M )+ (h,h,. . .), where the firstmajorizationis aconsequenceof (iii) (which appliesbecauseλi,ksatisfies(C2)),thesubsequentequalityisaconsequenceof(C2), which implies thatλi,k is notan eigenvalue of M (λ), and the last majorizationis immediate.

Now, (i),(ii)togetherwith(iv)–(vi)imply:

(a) r(M )≺ r(L)+ (h,h,. . .), (b) (M )≺ (L)+ (h,h,. . .),

(c) W (μ,L)≺ W (μ,M )+ (h,h,. . .),forallμ∈^C.

But (a)–(c)inturnimply thatM (λ)∈ O(L),accordingtoTheorem 7,andthis proves the“if” partofthestatement.

Now, let us provethe“only if” part,so let L(λ) begiven. Wefirst provethatB(L) containsO(ψc(L)),forallmapsψ :C→^C.

We aregoing tofirst provethatψc(L)∈ B(L). Tothis end,notethatB(L)=B(P · L(λ)· Q),foranyinvertiblematricesP,Q ofappropriatesize.Therefore,wecanassume thatL(λ) isgiveninKCFso,withoutlossofgenerality,L(λ) isoftheform

L(λ) = diag(Jλ1(λ), . . . , Jλs(λ), #L(λ)),