BOUNDS FOR THE COMPONENTWISE DISTANCE TO THE NEAREST SINGULAR MATRIX

Transcription

1 SIAM J. Matrix Aal. Appl. (SIMAX), 8():83 03, 997 BOUNDS FOR THE COMPONENTWISE DISTANCE TO THE NEAREST SINGULAR MATRIX S. M. RUMP Abstract. Th ormwis distac of a matrix A to th arst sigular matrix is wll kow to b qual to A /cod(a) for orms big subordiat to a vctor orm. Howvr, thr is o hop to fid a similar formula or v a simpl algorithm for computig th compotwis distac to th arst sigular matrix for gral matrics. This is bcaus Roh ad Poljak [7] showd that this is a NP -hard problm. Dot th miimum Baur-Skl coditio umbr achivabl by colum scalig by κ. Dmml [3] showd that κ is a lowr boud for th compotwis distac to th arst sigular matrix. I this papr w prov that κ is a uppr boud. This xtds ad provs a cojctur by N. J. Higham ad J. Dmml. W giv a xplicit st of xampls showig that a uppr boud caot b bttr tha κ. Asymptotically, w show that +l 2+ε κ is a valid uppr boud. Ky words. compotwis distac, NP-hardss, optimal Baur-Skl coditio umbr AMS subjct classificatios. 65F35, 5A60 0. Itroductio. Lt A b a by matrix ad dot its smallst sigular valu by σ (A). It is wll kow that th distac to th arst sigular matrix i th 2-orm or Frobius orm is qual to σ (A). Mor gral, for ay cosistt matrix orm big subordiat to a vctor orm w hav () mi { δa A + δa sigular } = A = A cod(a). A appropriat δa of rak ca b xplicitly calculatd (cf. [?], [3]). Such a prturbatio dos, i gral, altr ach compot of A. I may practical applicatios, o may b itrstd i lavig spcific compots such as systm zros ualtrd, for xampl, if th matrix ariss from som discrtisatio schm. Mor gral, this lads to th qustio of th compotwis distac to th arst sigular matrix. Th compotwis distac may b wightd by som ogativ matrix E. Mor prcisly, w dfi (2) σ(a, E) := mi { α IR A + Ẽ sigular whr Ẽij α E ij for all i, j }. If o such α xists, w st σ(a, E) :=. For sigular matrics, σ(a, E) = 0 for vry wight matrix E. Spcific valus of E ar E = A for rlativ prturbatios, or E = () for absolut prturbatios. Amog othrs, th compotwis distac to th arst sigular matrix was discussd i [8], [], [0], ad i [3]. I [8] w also fid a first approach towards a stimatio of th arss to sigularity i a orm ot big subordiat to a vctor orm, amly A := max A ij. i,j W caot xpct to fid a formula or v a simpl algorithm forq calculatig σ(a, E). This is bcaus Roh ad Poljak [7] provd that computatio of σ(a, E) is NP -complt. For a outli of thir proof s also [3]. Nvrthlss, w may fid bouds for σ(a, E), ad for classs of matrics v xplicit formulas. Aothr viw of σ(a, E) is th maximum valu such that th itrval matrix [A αe, A+αE] is osigular for α < σ(a, E). Th itrval matrix is dfid as big th st of all matrics à with A ij αe ij Ãij A ij + αe ij for all i, j, or i short otatio A αe à A + αe. Th itrval matrix is calld osigular if vry matrix à [A αe, A+αE] is osigular. I a vry itrstig papr [9], Roh gav 3 cssary ad sufficit critria for [A E, A + E] big osigular. Tchisch Iformatik III, TU Hamburg-Harburg, EißdorfrStraß 38, 207 Hamburg, Grmay

2 A thorough discussio of σ(a, E) ca b foud i th vry itrstig papr [3]. Dmml [3] provd that σ(a, E) is qual to th ivrs of mi κ(ad, ED), th miimum tak ovr all diagoal matrics D, whr κ(a, E) := A E dots th Baur-Skl coditio umbr. For ay p-orm, h provs mi D κ(ad, ED) = ρ( A E), xtdig a rsult by Baur []. I othr words, th miimum Baur-Skl coditio umbr achivabl by colum scalig is qual to th ivrs of ρ( A E). Dmml ad N. J. Higham cojctur that ρ( A E) ad σ(a, E) ar ot too far apart. Thy cojctur for rlativ prturbatios xistc of som costat γ IR, possibly dpdig o th dimsio, with (3) γ σ(a, A ) ρ( A A ). I this papr, our mai goal is to show xistc of such costats γ() ad to driv lowr ad uppr bouds for γ(). First, w show that σ(a, E) σ (A) for E 2 =. A corrspodig rsult for othr orms is giv i 2. Howvr, this boud ca b arbitrarily wak. Followig w giv som w bouds for σ(a, E). I 4 a prturbatio formula for dtrmiats is statd which is th ky to prov a uppr boud of γ(). I 5 w will prov γ. I 6, for arbitrary wight matrics E w prov (4) ρ( A E) σ(a, E) γ() ρ( A E) with γ() = c α for c = 2.4 ad α =.7. Morovr, for w show that for vry ε > 0, α ca b rplacd by +l 2+ε. I viw of γ, w cojctur γ =. I [3], Dmml gav rasos to b itrstd i th compotwis distac to th arst sigular matrix. I 2, w add a lowr ad uppr compotwis rror boud for th solutio of a liar systm Ax = b subjct to compotwis prturbatios of th matrix ad th right had sid. Such uppr bouds ar kow i th litratur ad ar valid for osigular A ad à A E with ρ( A E) <. W driv a compotwis boud for th miimum prturbatio of th solutio subjct to fiit prturbatios of A ad b. (4) shows that thos stimats covr prturbatio matrics à ot too far from th xt sigular matrix. Th papr is orgaizd as follows. I w itroduc th usd otatio. I 2 follows a compotwis lowr ad uppr prturbatio boud for fiit compotwis prturbatios of a liar systm. I 3, lowr bouds o σ(a, E) ar giv. For orthogoal matrics w show that γ (s (4)) is at last of th ordr of. I 4, a Shrma-Morriso-Woodbury lik prturbatio thorm for dtrmiats is giv. I fact, this is a quality for fiit prturbatios of a matrix. I 5 w driv uppr bouds o σ(a, E). For E big of rak, such as for absolut prturbatios, w show γ(), ad for rlativ prturbatios w giv a st of matrics A M (R) with γ() =. For a class of matrics icludig M-matrics wi prov γ() =, i.. σ(a, E) = ρ( A E). I 6 th rsults ar xtdd to obtai a xplicit uppr boud o γ() for gral A ad E, ad i 7 thos bouds ar quatifid ito (4). W clos with th cojctur that (4) is valid for γ() = for all A, E. If this is tru, th st of matrics giv i 5 would imply that iquality (4) with γ() = is sharp.. Notatio. I th followig w list som otatio from matrix thory, cf. for xampl [6], [5]. V (IR) dots th st of vctors with ral compots, M m, (IR) th st of ral m by matrics, ad M (IR) = M, (IR). Th compots of a matrix A M (IR) ar rfrrd by A ij or A i,j. For short otatio, compots of A ar rfrrd by A ij. () dots a vctor with all compots qual to, () M (IR) th matrix with all colums qual to (). Q k dots th st of strictly icrasig squcs of k itgrs chos from {,..., }. For ω Q k, w dot ω = (ω,..., ω k ). For C M (IR), ω Q k, C[ω] M k (IR) dots th k by k submatrix of C lyig i rows ad colums ω. A squc ζ = (i,..., i k ), k of mutually diffrt itgrs i ν {,..., } is 2

3 calld a cycl. W idtify th cycls (i,..., i k ) ad (i p,..., i k, i,..., i p ), whr p k. It is ζ := k. A full cycl ζ o {,..., } is a cycl ζ with ζ =. For C M (IR) ad a cycl ζ = (i,..., i k ) o {,..., }, w put Π ζ (C) := C ii 2... C ik i k C ik i, th cycl product for ζ. Not th last factor i th product. Thrfor, Π ζ (C) / ζ is th gomtric ma of th lmts of th cycl product. Each diagoal lmt C ii is a cycl product, amly of th cycl (i). (Hr our dfiitio diffrs from Egl/Schidr [4]). With o xcptio, throughout th papr, absolut valu ad compariso is usd compotwis. xampl, for A, B M (IR), For A B mas A ij B ij for i, j. Th xcptio ar cycls ζ = (i,..., i k ), whr ζ = k. Th sigular valus of a matrix A M (IR) ar dotd i dcrasig ordr with icrasig idics, i.. σ (A)... σ (A) 0. For A, E M (IR), E 0, σ(a, E) dots th compotwis distac, wightd by E, to th arst sigular matrix (cf. (0.2)). For fiit σ(a, E), th st of all matrics à M (IR) with à A σ(a, E) E is compact. For vry osigular à thr is a ighbourhood of à cosistig oly of osigular matrics. Thrfor σ(a, E) < δa M (IR) : δa = σ(a, E) E ad A + δa sigular, showig that w ar allowd to us a miimum i th dfiitio (0.2) of σ(a, E). ρ dots th spctral radius, whras ρ 0 dots th ral spctral radius: B M (IR) : ρ 0 (B) := max { λ λ IR is a igvalu of B }. If B has o ral igvalus, w st ρ 0 (B) := 0. I dots th idtity matrix of propr dimsio, spcially I k M k (IR) dots th k by k idtity matrix. A sigatur matrix S is a diagoal matrix with diagoal tris + or, i.. S = I. W frqutly us stadard rsults from matrix ad Prro-Frobius thory such as (5) A M k (IR), B M k (IR) Th st of ozro igvalus of AB ad BA ar idtical, cf. Thorm.3.20 i [5], ad A M (IR) ad A 0, x V (IR) with x > 0 (6) mi i (Ax) i x i ρ(a) max i (Ax) i x i. Th lattr ca b foud i [2]. 2. Fiit prturbatios for a liar systm. Calculatig bouds o σ(a, E) ca b motivatd, for xampl, by lookig at liar systms with fiit prturbatios of th iput data. For a liar systm Ax = b cosidr th prturbd systm à x = b with δa := à A, δb := b b, δx := x x. Th for osigular A, (7) A (I + A δa) ( x x) = à ( x x) = b Ãx = δb δa x. If ρ(a δa) <, th I + A δa ad à = A (I + A δa) ar osigular, ad (7) implis (8) δx = (I + A δa) A (δb δa x). 3

4 If ρ ( A A ) < th I A A is a M-matrix. If th prturbatios δa, δb ar compotwis boudd by δa A, δb b th (2.2) implis (9) δx (I A A) A ( b + A x ). For giv wight matrix A, cosidr th st of matrics with compotwis distac from A wightd by A ot gratr tha σ: à U σ (A, A) à A σ A. For σ ρ( A A), Prro-Frobius-Thory yilds ρ(i A Ã) = ρ( A (A Ã)) ρ( A A) <, ad thrfor rgularity of all à U σ(a, A). Th boud (2.3) rquirs A A to b covrgt, whras (8) is valid for ρ(a δa) <. Thrfor w may ask, how far a matrix à with ρ( A à A ) ca b from th arst sigular matrix. A aswr to this qustio shows how strog th assumptio ρ( A A) < is. (0) 3. Lowr bouds o σ(a, E). A simpl ad wll-kow lowr boud o σ(a, E) is ρ( A E) σ(a, E) for all osigular A M (IR), 0 E M (IR). This ca b s usig Prro-Frobius Thory ad ρ( A E) < ρ(a δa) < for all δa E A + δa = A (I + A δa) is osigular. Aothr lowr boud is (cf. [2], Thorm.8, p. 75) () σ (A) σ(a, E). σ (E) This ca b gralizd i th followig way. Thorm 3.. Lt b a matrix orm subordiat to a absolut vctor orm. Th for osigular A M (IR) ad 0 E M (IR), (2) A σ(a, E). E (2) is spcially valid for all p-orms. For absolut orms such as -orm ad -orm, (3) A E ρ( A σ(a, E), E) whras for th 2-orm (4) A 2 E 2 ρ( A E). Proof. To prov (2), lt δa M (IR) with δa σ E for σ < ( A E ). Th vctor orm is absolut implyig x = x ad x y x y for x, y V (IR) (cf. [3], Thorm II..2). Lt x V (IR) with x = ad δa = δa x. Th (5) δa = δa x = δa x σ E x σ E x = σ E < A. For vry 0 y V (IR) holds y A Ay, ad (5) yilds δa y δa y < A y Ay, ad thrfor (A + δa) y 0. 4

5 Thrfor, A + δa is osigular for δa σ E, ad σ < ( A E ), provig (2). For absolut matrix orms, ρ( A E) A E A E provig (3). For th 2-orm holds ρ( A E) A 2 E 2 A F E 2 = A F E 2 A 2 E 2, provig (4) ad th thorm. (3) shows that for absolut matrix orms such as th -orm or -orm, th boud (2) caot b bttr tha (0). Th 2-orm is ot absolut, ad (4) shows that th lowr boud () for σ(a, E) may b bttr up to a factor tha (0). I fact, w ca idtify a class of matrics for which this improvmt is approximatly achivd. Lt Q M (IR) b orthogoal, ad cosidr absolut prturbatios E = (). Th () yilds σ (Q) ( ) = σ(q, E). σ () O th othr had, E = () M (IR) ad x = () V (IR) imply { Q E } ( T ) x = E Q x = Q ij x, i,j ad (6) yilds ρ( Q E) = Q ij. If Q is a orthogoalizd radom matrix with compots uiformly i,j distributd i [, ], th Q ij /2. Thus, for th ratio btw th two lowr bouds () ad (0) w obtai { σ (Q)/σ (E) } / { /ρ( Q E) } 2 /2 =. Th sam huristic holds for E = Q, cf. [2]. For vry Hadamard matrix (H M (IR) with H T H = I) th ratio is qual to. This shds a first light o a possibl quatity γ() such that (4) holds. I 5 w will prov γ(). Exampl 3.2. Th lowr boud () may b arbitrarily wak. Cosidr ( ) 2ε ε A = ad E = A for som ε > 0. ε A is a diagoally domiat M-matrix. As w will s i (5.5), A big M-matrix implis quality i (0), i.. σ(a, A ) = ρ( A A ) = + 0( ε). O th othr had, σ 2 (A)/σ ( A ) = 2ε + 0(ε 2 ) udrstimats σ(a, A ) arbitrarily. This corrspods to σ 2 (A) = 2ε + 0(ε 2 ). That mas, th ormwis distac i th 2- orm or Frobius orm to th arst sigular matrix ca b arbitrarily small compard to a compotwis distac. 4. A prturbatio thorm for dtrmiats. A lowr boud o σ(a, E) is obtaid by provig rgularity of a st of matrics. This was do i 3 by usig spctral proprtis. To obtai a uppr boud o σ(a, E), w may costruct a spcific prturbatio δa with δa σ 0 E, σ 0 IR such that A+δA is sigular. This provs σ(a, E) σ 0. Aothr possibility to obtai a uppr boud o σ(a, E) is th followig. If δa σ 0 E ad dt(a) dt(a + δa) 0, th a cotiuity argumt yilds σ(a, E) σ 0. Thrfor w stat th followig xplicit formula for th rlativ chag of th dtrmiat of a matrix subjct to a rak-k prturbatio. It is a Shrma-Morriso-Woodbury lik prturbatio formula for dtrmiats. Lmma 4.. Lt A M (IR) ad U, V M,k (IR) b giv. Th for osigular A, (6) dt(a + UV T ) = dt(a) dt(i k + V T A U), whr I k dots th k by k idtity matrix. 5

6 Proof. It is dt(a + UV T ) = dt(a) dt(i + A UV T ). Dotig th igvalus of X M (IR) by λ i (X) implis dt(i + A UV T ) = λ i (I + A UV T { ) = + λi (A UV T ) }. i= i= Th st of ozro igvalus of A UV T ad V T A U ar idtical (s (5)), thus provig th lmma. This lmma has a ic ad for itslf itrstig corollary. Corollary 4.2. Lt A M (IR) ad u, v V (IR). Th for osigular A, (7) dt(a + uv T ) = dt(a) ( + v T A u). For arbitrary A M (IR) holds (adj(a) dots th adjoit of A), (8) dt(a + uv T ) = dt(a) + v T adj(a) u. Th corollary shows that th rlativ chag of th dtrmiat is liar for rak- prturbatios of th matrix. Th scod wll-kow formula follows, for xampl, by a cotiuity argumt usig A adj(a) = dt(a) I. 5. Uppr bouds o σ(a, E). Th prturbatio lmma for dtrmiats giv i 4 allows for othr lowr bouds o σ(a, E). Th first rsult ca b foud i [8], Corollary 5., (iii). Thorm 5.. Lt A M (IR) b osigular ad E M (IR) with E 0. Th (9) σ(a, E) { max A E } i ii, whr 0 is itrprtd as. { Proof. St α := max A E } 0 ad lt i b a idx, for which this maximum is achivd. Dot i ii th iν-th compot of A by A iν (20) T i A u = α ν= ad dfi u V (IR) by u ν := α sig(a iν ) E νi. Th A iν E νi =, ad Corollary 4.2 implis dt(a + u T i ) = 0. Now ut i α E yilds σ(a, E) α. Exampl 5.2. Th uppr boud (9) ca b arbitrarily wak. Cosidr ε 0 /ε /ε 0 ε (2) A = ε 0, E = A with /ε /ε A A /ε /ε 0 ε /ε /ε whr th compots of A A ar accurat up to a rlativ rror ε. Th (9) givs σ(a, A ) +0(ε). O th othr had, dt(a + ε δa) = 0 for δa = 0 0 0, showig σ(a, A ) ε. 6,

7 I Thorm 5., a rak- prturbatio is usd to prov (9). I a ormwis ss, th miimum distac to th arst sigular matrix is achivd by a rak- prturbatio. This is o logr tru for compotwis distacs, as will b show by th followig xampl. Exampl 5.3. Accordig to Corollary 4.2, th smallst σ such A + σ is sigular with σ A ad rak() = is giv by σ = ϕ, whr ϕ is a optimal valu of th costrait optimizatio problm ϕ(u, v) := v T A u = Mi! subjct to uv T A. I Exampl 5.2, partitio th vctors u, v V 4 (IR) ito two vctors U i, V i V 2 (IR), i {, 2}, ithr havig 2 compots. That mas u = (U, U 2 ) T, v = (V, V 2 ) T. Lt uv T A, i.. U i V T i ε I ad U i V T j () 22 for i, j 2, i j. Th larg lmts of A ar i th uppr lft ad lowr right 2 by 2 block: ( ) ( ) ( A X Y with X = Y X 2ε ad Y = 4 up to a rlativ rror of th ordr ε. Thrfor v T A u V T XU + V T 2 XU 2 + V T Y U 2 + V T 2 Y U 2ε X ij + 2 Y ij 6 Thrfor, Corollary 4.2 implis that th miimum distac to th arst sigular matrix subjct to rak- prturbatios wightd by A is at last /6 compard to σ(a, A ) ε. This obsrvatio shds light o th difficultis to calculat σ(a, E) or to fid uppr bouds for it. O may dfi th rak-k compotwis distac to th arst sigular matrix as follows σ k (A, E) := mi{ α IR A + Ẽ sigular for Ẽ α E ad rak(ẽ) k }. W us rak(ẽ) k bcaus E may b rak-dficit. W hav just s i Exampl 5.3 that σ 2(A, E)/σ (A, E) may b arbitrarily small. Giv th lowr boud (0), o may ask whthr thr xist fiit costats γ() IR oly dpdig o such that (22) ρ( A E) σ(a, E) γ() ρ( A E) for all osigular A M (IR) ad 0 E M (IR). This qustio has b raisd i [3] ad aswrd for som classs of matrics. Th mai purpos of this papr is to driv bouds for γ(). This will b do by usig Lmma 4.. For this purpos w d th followig rsult by Roh (for otatio s ). Thorm 5.4. (Roh) For osigular A M (IR) ad 0 E M (IR) holds max S,S 2 ρ 0 (S A = σ(a, E), S 2 E) ) whr ρ 0 dots th ral spctral radius ad th maximum is tak ovr all sigatur matrics. itrprtd as. /0 is Proof. cf. [9]. W start with a thorm boudig γ() for gral wight matrics E, ad idtify a class of matrics with γ() =. Thorm 5.5. For osigular A M (IR) ad 0 E M (IR), th followig is tru. i) Assum a matrix S M (IR) of rak xists with 7

8 + if A ij > 0 S ij = if A ij < 0. + or if A ij = 0 Th (22) holds with γ() =. ii) If 0 < η E ij ζ for all i, j, th (22) holds with γ() = ζ/η. Proof. Lt S = uv T with u, v V (IR), u = v = (). Dfiig S = diag(u), S 2 = diag(v), w hav S A S 2 = A ad Roh s charactrizatio i Thorm 5.4 provs th first part. W.l.o.g. assum σ(a, E) <. It is η A max ( A E) ii ad ρ( A E) A E ζ A. i Thus, Thorm 5. provs th scod part ad thrfor th thorm. For importat classs of matrics such as ogativ ivrtibl matrics, amog thm all M-matrics, w alrady hav a prcis formula for σ(a, E): (23) A M IR ogativ ivrtibl, 0 E M (IR) σ(a, E) = ρ( A E). Exampl 5.6. If costats γ() with (22) xist at all, w ca giv a lowr boud o γ() by mas of th followig. Dfi A M (IR) by s 0 A := (24) with s := ( ) Th dtrmiat of A calculats to dt(a) = A ii + ( ) + Π ζ (A) = 2, whr ζ = (,..., ) i= ad ζ (A) = A 2 A A, A. If th lmts of A ar afflictd with rlativ prturbatios, i.. E = A, th oly th s ad s chag. Thrfor, ay à with à A σ A with σ < is osigular, ad thrfor σ(a, A ) =. O th othr had, A A = () ad ρ( A A ) =. This provs th followig lmma. Lmma 5.7. If costats γ() IR with (22) for vry osigular A M (IR) ad 0 E M (IR) xist at all, th γ(). Nxt w show that γ() for E big of rak. For th proof w us Corollary 4.2, which is a cosquc of Lmma 4. for k =. I th rmaiig part of th papr, w will xtd this proof to k > to obtai uppr bouds for γ() ad for gral A, E. Thorm 5.8. Lt osigular A M (IR) ad 0 E M (IR) with E = uv T u, v 0. Th ρ( A E) σ(a, E) ρ( A E). Proof. Accordig to Thorm 5.4 ad usig (.), for som u, v V (IR), (25) σ(a, E) = max S,S 2 ρ 0 (S A S 2 uv T ) = max S,S 2 v T S A S 2 u, whr th maximum is tak ovr all sigatur matrics S, S 2. For ay i, i, w ca choos appropriat sigatur matrics S, S 2 such that v T S A S 2 u v i ( A u) i. Usig (25) this yilds σ(a, E) max i O th othr had, usig (.), v i ( A u) i. 8

9 ρ( A E) = ρ( A uv T ) = v T A u max i v i ( A u) i. Corollary 5.9. For osigular A M (IR) ad absolut prturbatios, i.. E = (), stimatio (22) holds with γ() =. 6. Estimatio of γ(). To mak furthr progrss i th stimatio of γ() w show that for osigular A, σ(a, E) dpds cotiuously o A ad E. Usig this w ca rstrict th class of matrics A ad E to matrics with oly ozro compots. For th proof w ca hardly us a simpl cotiuity argumt o ρ 0 (S A S 2 E) i coctio with Thorm 5.4. This is bcaus th sarch domai is rstrictd by E, ad th (i absolut valu) largst ral igvalu may b multipl ad bcom complx udr arbitrarily small prturbatios. Lmma 6.. For osigular A M (IR), σ(a, E) dpds cotiuously o A ad E. Proof. For σ(a, E) = w show that σ(ã, Ẽ) bcoms uboudd for à A, Ẽ E. A compactss ad cotiuity argumt shows that for vry fiit 0 < c IR: c E : dt(a + ) δ > 0. For vry Ã, Ẽ clos ough to A, E, this implis dt(ã + ẽ) δ/2 > 0 for vry ẽ c Ẽ, ad hc σ(ã, Ẽ) > c. Assum σ := σ(a, E) <. W will show that for small ough ε > 0, thr xists som δ > 0 such that both of th followig statmts ar tru: (26) (27) M (IR) : (σ ε) E dt(a) dt(a + ) > δ, M (IR) : (σ + ε) E ad dt(a) dt(a + ) < δ. (26) is s as follows. For ε > 0, th st of matrics A + with (σ ε) E is ompty ad compact. Hc, dt(a) dt(a + ) achivs a miimum o this st. By dfiitio of σ, this miimum is positiv. To s (27), obsrv that dt(a) dt(a + ) 0 for all σ E. For ay idx pair i, j, th dtrmiat dt(a + ε i T j ) dpds liarly o ε. Now procd as follows. Thr is som such that A + is sigular ad = E. If for a idx pair i, j, th dtrmiat dt(a + ) is idpdt o ij, th rplac ij by 0. At ach stp of this procss, dt(a + ) = 0 ad E. Th dfiitio of σ(a, E) < implis that durig this procss w must arriv at som ad a idx pair k, l, such that dt(a + ) is ot costat wh chagig kl. Th dfiig M (IR) by ij := ij for (i, j) (k, l) ad kl := kl ( + ε ) for small ε > 0 provs (27). Now th cotiuity of th dtrmiat implis for Ã, Ẽ clos ough to A, E, ẽ (σ ε) Ẽ : dt(ã) dt(ã + ẽ) > δ/2 ad ẽ (σ + ε) Ẽ : dt(ã) dt(ã + ẽ) < δ/2, ad thrfor σ(a, E) ε < σ(ã, Ẽ) < σ(a, E) + ε. Corollary 6.2. If (22) holds for ach E > 0, th it holds for ach E 0. Our goal for this chaptr is to prov th followig uppr boud for σ(a, E). Th quatitis ϕ t occurig i this stimatio will b quatifid ad stimatd i 7. Propositio 6.3. Lt A, E M (IR) with A osigular ad E 0 b giv. Dfi rcursivly ϕ :=, ϕ 2 := ad ϕ t IR, 2 < t IN to b th (uiqu) positiv root of (28) t P t (x) IR[x] with P t (x) := x t x t 2 ϕ ν ν x t ν. ν= 9

10 Th (29) σ(a, E) ϕ ρ( A E). Thrfor, th quatitis γ() dfid i (5.4) satisfy (30) γ() =, γ(2) = 2 ad γ() ϕ. Th proof divids ito svral parts ad ds som prparatory lmmata. First, w will costruct a spcific rak-k prturbatio i ordr to b abl to apply Lmma 4. to boud γ() for gral A, E. W us th sam pricipl as i th proof of Thorm 5. adaptd to rak-k prturbatios. Lmma 6.4. Lt osigular A M (IR) ad 0 E M (IR) b giv, ad st C := A E. For k dfi { i i + for i < k (3) := for i = k ad U, V M,k (IR) by U νi := sig(a iν ) E νi ad V µi := δ µi for µ, ν, i k ad th Krockr symbol δ. St C := V T A U. Th i) C C[ω] for ω = (,..., k). ii) C ii = C ii for i k. iii) UV T E. iv) σ(a, E) {ρ 0 ( C)}, whr 0 is itrprtd as. Proof. For i, j k follows (V T A U) ij = ν= µ= V µi A µν U νj ad thrfor C C[ω] ad i). For i k holds C ii = (V T A U) ii = ν= µ= ad thrfor ii). For µ, ν holds (UV T ) νµ = k U νi V µi, i= ν= V µi A µν U νi = A iν E νj = C ij, ν= A iν sig(a iν ) E νi = C ii. such that (UV T ) νµ = E νµ for µ k, ad (UV T ) νµ = 0 for k + µ. This provs iii). For λ := ρ 0 ( C) > 0, it is dt(λ I s C) = 0 for s = or s =. Lmma 4. implis dt(a s λ UV T ) = dt(a) dt(i k s λ V T A U) = 0. Togthr with iii) ad th dfiitio (0.2) of σ(a, E), this provs iv) ad th thorm. Our aim is to costruct a rak-k prturbatio of A with larg ral spctral radius. Th Lmma 4. allows to giv a uppr boud o σ(a, E). A first stp is th followig, first gralizatio of Thorm 5.. It will latr yild th prcis valu for γ(2). Thorm 6.5. Lt A M (IR) b osigular ad E M (IR) with E 0. For C := A E holds (32) σ(a, E) max i,j. Cij C ji 0

11 Proof. For i = j, (32) has b provd i Thorm 5.. Rordrig of idics puts th cycl (i, j), for which th maximum i (6.7) is achivd, ito th cycl (,2), ad Lmma 6.4 provs for i j xistc of a 2 by 2 matrix C = ( α β γ δ) with 0 β = Cij, 0 γ = C ji, α C ii, δ C jj, ad σ(a, E) ρ 0 ( C). If αδ βγ, th C ii C jj C ij C ji ad Thorm 5. yilds (32). Othrwis, dt( C) < 0. Th charactristic polyomial of C is λ 2 trac( C) λ + dt( C), so that th igvalus of C { ar 2 trac( C) ± trac( C) 2 4 dt( C) } { = 2 α + δ ± } (α δ) 2 + 4βγ ar both ral. Th absolut valu of o of thm is ot lss tha βγ, i.. σ(a, E) ρ 0 ( C) (βγ) /2. Th ida of th proof of Thorm 6.5 is th followig: for a giv cycl of C of lgth 2, a suitabl rak-2 prturbatio of A is costructd which allows to prov a uppr boud of σ(a, E) by usig Lmma 6.4. I th followig w will carry this ida to cycls of C of lgth k, k. First, w will idtify a class of matrics for which w ca giv xplicit lowr bouds for thir ral spctral radius. Th class of matrics is costructd i such a way that th matrics giv i Lmma 6.4 ca b usd to boud σ(a, E) from abov. Lmma 6.6. Lt ogativ C M k (IR) ad som 0 < a IR b giv. Dfi ϕ :=, ϕ 2 :=, ad for t > 2 dfi rcursivly ϕ t IR to b th positiv zro of t (33) P t (x) IR[x] with P t (x) := x t x t 2 ϕ ν ν x t ν. Suppos ν= (34) µ < k ω Γ µk : (Π ω (C)) /µ ϕ µ a, ad for ω = (,..., k), (35) Π ω (C) /k ϕ k a. Th, for i dfid as i (6.6) ad vry C M k (IR) with (36) C C ad C ii = C ii for i k, holds ρ 0 ( C) a. Proof. Th proof divids i th followig parts. First, w trasform C ito a stadard form such that all C ii i th cycl (,..., k) i (35) ar qual. Scod, w boud C by a circulat, show rgularity of that matrix ad dt( C λi) 0 for all 0 λ < a. Fially, th sig of th dtrmiat of ay C with (36) is dtrmid, from which th lmma follows. Th cas k = is trivial; for k = 2 th proof of ρ 0 ( C) a is icludd i th proof of Thorm 6.5. Assum k > 2, ad st b := Π ω (C) /k. Dirct computatio shows that ay similarity trasformatio of C by a diagoal matrix D lavs all cycl products ivariat. Thus (34) ad (35) rmai valid for ay diagoal D with positiv diagoal tris. Dfi diagoal D M k (IR) by f i := b C ii ad D ii := k f ν for i k. ν=i W show that w.l.o.g. C ca b rplacd by D CD. W hav f i > 0, ad (6.0) implis D =. It is ( k ) ( k ) (37) (D CD) ii = fν C ii f ν = C ii f i = b. ν=i ν=i

12 If C Mk (IR) is ay matrix satisfyig (36), th D CD D CD, ad (37) yilds (D CD)ii = (D CD) ii = b. Sic th st of igvalus of C ad D CD ar idtical, w ca rstrict our atttio to matrics C M (IR), C 0 ad (38) St (39) C ii = b for i k. C = c, b c,k... c,2 c 2,2 c 2, b c 2,k... c 2,3... c 3, b... c k,k c k,k 2... b b c 0,k... c 0,. Lt µ IN, µ < k b giv ad dfi ω Γ µk by ω = (,..., µ). Th sttig q := a/b, (34) implis c µµ µ i= b (ϕ µ a) µ ad thrfor c µµ b ϕ µ µ q µ. Applyig th sam argumt succssivly for ω = ( i, (i + ) mod µ,..., (i + µ) mod µ ) yilds (40) c i,µ b ϕ µ µ q µ for all 0 i < k, µ < k. Thrfor, (4) C b c c k c 2 c 2 c c k... c 3 c c k c k 2... c k c =: b C with c µ := ϕ µ µ q µ for µ <. Lt C M k (IR) with (36) b giv, ad lt λ IR with 0 λ < a. Nxt w show that all matrics C λi ar osigular. By assumptio (6.) ad usig (6.6), (42) C λi C + λ I b C + λi ad ( C λi) ii = C ii = C ii = b. By (4) ad (33), usig q := a/b ϕ k from (35) ad ϕ 2 =, w hav for k 3, { } { } λ + b k c ν < b q + k ϕ ν ν q ν b ϕ k + k ϕ ν ν ϕ ν k = (43) ν= { = b ϕ k+ k ϕ k 2 k ν= + k ν= ν= } ϕ ν ν ϕ k ν k = b ϕ k+ k ϕ k k = b. This shows that th lmt b = C ii = C ii strictly domiats th sum of th absolut valus of th othr compots i ach row of C + λi ad of C λi. That mas, multiplicatio by a suitabl prmutatio matrix producs a strictly diagoally domiat matrix ad provs rgularity of vry C λi with C satisfyig (36) ad 0 λ < a. W provd that for vry C M k (IR) with (36), th dtrmiat of C λi is ozro for 0 λ < a. Thrfor, th valu of th charactristic polyomial p(λ) = dt(λi C) of C has th sam sig for 0 λ < a. Now p(λ) + for λ +. Thrfor, th lmma is provd if w ca show p(0) < 0, bcaus i this cas th charactristic polyomial must itrsct with th ral axis for som λ a, thus provig ρ 0 ( C) λ a. W alrady provd that vry matrix C satisfyig (36) is osigular. Thrfor 2

13 sig ( p(0) ) = sig ( dt( B) ) for vry matrix B with B C ad B ii = C ii = b. Dfi { C ii for j = i B ij :=. 0 othrwis Th sig ( dt(b) ) = ( ) k+ ad thrfor sig ( p(0) ) = ( ) 2k+ =. Th thorm is provd. Exampl 6.7. O ca show that, at last for odd, th bouds i Lmma 6.6 ar sharp i th ss that thr ar xampls with quality i (34) ad (35) such that C with (36) xists with ρ 0 ( C) = a. Cosidr a b c a b c C := c a b with b := ϕ 3 a ad c := a/ϕ 3 ad C := c a b. b c a b c a Th C = ϕ a = a, C 2 C 2 = ϕ 2 a = a ad (C 2 C 23 C 3 ) /3 = ϕ 3 a. C is a circulat, ad its igvalus comput to P (ε k ), k = 0,, 2 whr ε = 2πi/3 ad P (x) = bx 2 cx a (cf.[6]). It is b c a = b( ϕ 2 2q 2 ϕ q) = b ϕ 3 = a with q := a/b. Th othr two igvalus ar complx, thus ρ 0 ( C) = a. Th xampl xtds to odd IN. Th combiatio of Lmma 6.4, Thorm 6.5, ad Lmma 6.6 givs th ky to costruct a rak-k prturbatio of A to achiv a uppr boud for σ(a, E). Th followig thorm is th gralizatio of Thorms 5. ad 6.5 for cycls of lgth k, k. Thorm 6.8. Lt A, E M (IR) with osigular A ad E 0 b giv ad dfi C := A E. For k ad ay ω Γ k st (44) 0 τ := ( Π ω (C) ) /k. Th for ϕ k dfid as i Lmma 6.6, σ(a, E) ϕ k /τ. I othr words, ϕ k dividd by th gomtric ma of th lmts of ay cycl of C bouds σ(a, E) from abov. Proof. Lt som ω Γ k ad τ from (44) b giv ad st a := τ/ϕ k. If for k = or k = 2 thr xists som ω Γ k with {Π ω (C)} /k ϕ k a, th ϕ = ϕ 2 = ad Thorm 5. ad Thorm 6.5 imply σ(a, E) a = ϕ k /τ. Thrfor, w may assum {Π ω (C)} /k < ϕ k a for all ω Γ k ad k {, 2}. Hc, thr is som m IN, 2 m k such that ad µ < m ω Γ µ : (Π ω (C)) /µ ϕ µ a, ω Γ m : (Π ω (C)) /m ϕ m a. Aftr suitabl rarragmt of idics w may assum ω = (,..., m), ad Lmma 6.4 yilds a matrix C M m (IR) with proprtis i) ad ii) of Lmma 6.4 ad σ(a, E) ρ 0 ( C). But Lmma 6.6 shows for all such matrics ρ 0 ( C) a = τ/ϕ m. Rgardig m k, th thorm is provd if w ca show (45) t IN ϕ t ϕ t+. W kow ϕ = ϕ 2 =, from th dfiitio (6.8) w s ϕ 3 = + 2, ad for t 3 P t+ (x) = x P t (x) ϕ t t. Hc P t+ (ϕ t ) < 0 ad ϕ t+ > ϕ t. Th thorm is provd. Thorm 6.8 rducs th problm of fidig uppr bouds of σ(a, E) to fidig propr cycls of som lgth k of A E with larg gomtric ma corrspodig to a suitabl rak-k prturbatio. This is do i th followig proof of Propositio

14 Proof. of Propositio 6.3. Corollary 6.2 allows us to assum E > 0. Thrfor, A E is positiv, ad Prro-Frobius Thory yilds xistc of a positiv igvctor x V (IR) with A E x = ρ( A E) x, ρ( A E) > 0. Dfi th diagoal matrix D x M (IR) by (D x ) ii := x i. W may rplac A by A D x ad E by E D x, bcaus for ay osigular diagoal matrics D, D 2, σ(a, E) = σ(d AD 2, D ED 2 ). This is bcaus δa σ E iff D δa D 2 σ D ED 2 ad A + δa is sigular iff D AD 2 + D δa D 2 is sigular (cf. [3]). Th C := (A D x ) E D x = D x A E D x ad C () = ρ( A E) (). That mas C is a multipl of a row stochastic matrix. St ρ := ρ( A E). Dot a idx of th maximal compot of C i row i by m i. Th ithr { m i i } = {,..., } or, thr is a cycl m j, m j+,..., m j+k, m j+k = m j of lgth k. That mas, with a suitabl rumbrig, thr is som k IN, k such that for th uppr lft k by k pricipal submatrix of C holds (46) C ii ρ/ for i k, whr i is dfid as i (6.6). Th Thorm 6.8, (6.20) ad (6.2) imply for ω = (,..., k), σ(a, E) ϕ k {Π ω (C)} /k ϕ k /ρ ϕ /ρ. I th rmaiig 7, w will rplac th boud (30) by givig xplicit bouds for γ() oly dpdig o. A asymptotic boud will b giv as wll. 7. Explicit bouds for γ(). Th mai rsult i 6 is th uppr boud (30) i Propositio 6.3. This boud is giv i trms of ϕ k, th positiv zros of th polyomial P t dfid i (28). I th rmaiig part of th papr w will giv bouds o γ() showig th dpdc o by a simpl fuctio. Morovr, th asymptotic bhaviour of γ() for is giv. Th polyomials P t (x) IR[x] dfid i (28) satisfy t (47) P t (x) = x t x t 2 ϕ ν ν x t ν ad P t (ϕ t ) = 0 for t > 2. Thrfor, for 3, (48) ϕ + i= By (6.20), x + (49) x + i= i= ϕ i i ϕ i =. ν= ϕ i i xi is strictly icrasig for x > 0. Hc, for x > 0, ϕ i i x i implis x ϕ, that is ϕ x. W ar aimig o a boud of th form (50) ϕ k c k α for som costats c ad α. To dtrmi c ad α, w otic that if (7.4) is satisfid for k <, th ( ) αi i (5) c α implis ϕ c α. i= This is bcaus th lft had sid of (7.5) yilds c α + i αi αi (c α ) + ϕ i i (c α ) i, i= ad (7.3) implis (c α ) ϕ. i= 4

15 Thrfor, our first stp is to driv uppr bouds for ( ) iα i (52) σ i with σ i :=. i= σ i dpds o ad α. W us th abbrviatio σ i for fixd ad α ad omit xtra paramtrs for bttr radability. I ordr to stimat th sum (52), w will split it ito 3 parts, which will b boudd idividually. For i holds σ i+ σ i = ad thrfor (53) σ i < ( i + ) (i+)α ( ) (i+)α i ( ) α σi+ for i. i For all β IR with < β < ad k := β i=k ad thrfor 2 σ i < σ + σ σ k > ad k β yilds i=k+ i=k ( ) α i = ( ) α σi+ = σ + i { ( σ σ k > ( β α ) (i ) i=k+ σ i. ( ) α ( i + ) (i+)α > i holds k < β k. Th (53) givs ) α } σ i i=k+ i=k+ ( ) α σ i, (i ) { ( ) α } σ i, k By choic, β >, ad α 0 implis ( β α ) >. Thrfor, ( ) ( )α (54) σ i < ( β α ) σ = ( β α ) =: µ i=k ( ) α i α, holds for vry α 0, < β < ad k := β. This is th first part of th sum (7.6) for a suitabl k to b dtrmid. Dfi ( x ) xα ( x ) xα f(x) := with f (x) = {α l x } + α. For x > 0, f(x) has xactly o miimum at x =. Th f(i) = σ i shows (55) σ k σ l for k l, ad σ k σ l for St M := /. Th k = β (56) k i=m k l. satisfis M k, ad (55) implis for 3, ( β σ i (k M) σ k < + ) ( ) β f < β ( ) β f ( ) βα β + =: ν. This is th scod part of th sum (7.6). Fially, (7.9) implis (57) M i= σ i < ( ) α + ( ) 2α 2 + ( ) 3α 3 + ( ) 4α 4 ( ) =: ξ, which is th third part of th sum (7.6). Th iqualitis (54), (56) ad (57) togthr yild (58) α + i= ( ) iα i α + µ + ν + ξ for 3. 5

16 Nxt, w show that all thr squcs µ, ν, ξ ar dcrasig for larg ough. ( + ) is mootoically icrasig for, thrfor for 2, ( ) α ( ) ( )α + µ + µ. Suppos (59) 0 Th for 0, + ( β { ( ) βα }. β ) βα ( + ) ( ) βα β ( + ) ( ) (+) βα β + ( β α ) βα +, ad thrfor ν + ν for 0 with 0 satisfyig (59). Fially, for ad α > 0.25, 4α < 0 ad thrfor ( ) 4α ( ) ( ) 4α ( ) ( + ) 4α 4α ξ + ξ. + Summarizig, this provs th followig lmma. Lmma 7.. Dfi ϕ :=, ϕ 2 := ad rcursivly ϕ to b th positiv zro of P (x) giv i (47). Lt costats c, α IR, α l 2 ad 3 0 IN b giv with ϕ c α for < 0. If a costat β IR, < β < xists such that (59) is satisfid ad µ, ν, ξ dfid i (54), (56), (57) satisfy (60) α + µ + ν + ξ for = 0, th ϕ c α for all IN. Proof. (50) is satisfid for k <, ad (58) ad (7.4) prov th lft had sid of (5) for = 0, ad thrfor (7.4) for k =. Th quatitis α, µ, ν ad ξ ar dcrasig for icrasig. Thus, (7.4) ad thrfor (7.4) is valid for all 0. By assumptio, ϕ c α for < 0 as wll. For xampl, for β := 2.697, α := 0.7 ad 0 := 3000, o chcks by xplicit calculatio ϕ 2.32 α for 0. Th lowr boud (59) for 0 is lss tha 83, µ < 0.992, ν < , ξ < , ad α < for = 0. This provs th followig rsult. Corollary 7.2. For all, ϕ Th diffrc ϕ is lss tha 2.8 for < 20, ad lss tha 2.0 for Summarizig, Corollary 7.2, Propositio 6.3, ad Lmma 5.7 prov th followig rsult. Propositio 7.3. Lt A, E M (IR) with osigular A ad E 0 b giv. Th for all with ρ( A E) σ(a, E) γ() ρ( A E), γ() Th lowr boud for γ() is sharp. Fially, w will show th asymptotic bhaviour of uppr bouds for γ(). Lt α := l(2 + 2η), η > 0. For ay < β < ad, α 0, µ ( β α ) α, ν 0 ad ξ 0. 6

17 For l β := (2 + η)/(2 + 2η), a short computatio yilds ( β α ) α = 2 + η 2 + 4η + 2η 2 <. Hc, for this β ad larg ough 0, (59) holds ad α + µ + ν + ξ < for all 0. Thrfor, for larg ough c with ϕ c α for < 0, Lmma 7. implis that ϕ c α for all IN. Usig α > l 2 provs th followig. Propositio 7.4. Lt γ() b dfid as follows: γ() := if{ σ(a, E) ρ( A E) A M (IR) osigular ad 0 E M (IR) }. Th γ() is fiit for all IN. Morovr, for ay ε > 0 thr xists som 0 IN such that for all 0 holds (6) γ() +l 2+ε. Th lowr boud i (6) is sharp. ) I his papr [3], Dmml showd that for th Baur-Skl coditio umbr κ(a, E) := A E with ay p-orm, p, thr holds ρ( A E) =, mi κ(ad, ED) D whr th miimum is tak ovr all diagoal D. Thus, Propositio 7.3 ad Propositio 7.4 prov that th compotwis rlativ distac to th arst sigular matrix for ay wight matrix E 0 is ot too far from th rciprocal of th smallst coditio umbr achivabl by colum scalig. Th vidc prstd i this papr lads us to th followig cojctur. Cojctur 7.5. For all osigular A M (IR) ad 0 E M (IR) holds (62) ρ( A E) σ(a, E) ρ( A E). If th cojctur is tru, Lmma 5.7 shows that it is sharp. Ackowldgmts. Th author wishs to thak Jiri Roh for may hlpful commts. REFERENCES [] F. Baur, Optimally scald matrics, Numrisch Mathmatik 5, (963), pp [2] L. Collatz, Eischlißugssatz für di charaktristisch Zahl vo Matriz, Math. Z., 48 (942), pp [3] J. Dmml, Th Compotwis Distac to th Narst Sigular Matrix, SIAM J. Matrix Aal. Appl., 3 (992), pp [4] G. Egl ad H. Schidr, Th Hadamard-Fischr Iquality for a Class of Matrics Dfid by Eigvalu Mootoicity, Liar ad Multiliar Algbra 4, (976), pp [5] R. Hor ad C. Johso, Matrix Aalysis, Cambridg Uivrsity Prss, 985. [6] M. Marcus ad H. Mic, A survy of matrix thory ad matrix iqualitis, Dovr publicatios, Nw York, 992. [7] S. Poljak ad J. Roh, Chckig Robust Nosigularity Is NP-Hard, Math. of Cotrol, Sigals, ad Systms 6, (993), pp. 9. [8] J. Roh, Narss of Matrics to Sigularity, KAM Sris o Discrt Mathmatics ad Combiatoris, (988). [9], Systms of Liar Itrval Equatios, Liar Algbra Appl. 26, (989), pp [0], Itrval Matrics: Sigularity ad Ral Eigvalus, SIAM J. Matrix Aal. Appl. 4, (993), pp [] S. Rump, Estimatio of th Ssitivity of Liar ad Noliar Algbraic Problms, Liar Algbra ad its Applicatios 53, (99), pp. 34. [2], Vrificatio Mthods for Ds ad Spars Systms of Equatios, i Topics i Validatd Computatios Studis i Computatioal Mathmatics, J. Hrzbrgr, d., Elsvir, Amstrdam, 994, pp ) Not addd i proof: I th matim it has b show by th author that γ() ( ). 7

18 [3] G. Stwart ad J. Su, Matrix Prturbatio Thory, Acadmic Prss,