Localization Theorems for Matrices and Bounds for the Zeros of Polynomials over Quaternion Division Algebra

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1 Floma 32:2 28) hps://doorg/2298/fil82553a Publshed by Faculy of Scences and Mahemacs Unversy of Nš Serba Avalable a: hp://wwwpmfnacrs/floma Localzaon Theorems for Marces and Bounds for he Zeros of Polynomals over Quaernon Dvson Algebra Sk Safue Ahmad a Iskhar Al a a Dscplne of Mahemacs School of Basc Scences Indan Insue of Technology Indore Smrol Indore Inda Absrac In hs paper we derve Osrowsk and Brauer ype heorems for he lef and rgh egenvalues of a uaernonc marx Generalzaons of Gerschgorn ype heorems are dscussed for he lef and he rgh egenvalues of a uaernonc marx Afer ha a suffcen condon for he sably of a uaernonc marx s gven ha generalzes he sably condon for a complex marx Fnally a characerzaon of bounds s derved for he zeros of uaernonc polynomals Inroducon Quaernons are exensvely used n he programmng of vdeo games compuer graphcs uanum physcs flgh dynamcs and conrol heory ec The soluons of lnear dfferenal euaons wh uaernon consan coeffcens lead o uaernonc polynomals So he sably analyss of such dfferenal euaons can be suded hrough localzaon heorems of uaernonc marces In recen pas fndng he zeros of uaernonc polynomals and fndng he bounds of zeros of uaernonc polynomals have ganed much aenon n he leraure Ths paper aemps o sudy he localzaon heorems for marces over a uaernon dvson algebra whch ncludes he Osrowsk Brauer and Gerschgorn ype of heorems Bounds for he zeros of uaernonc polynomals are also consdered Localzaon heorems for uaernonc marces have receved much aenon n he leraure due o her numerous applcaons n pure and appled scences; see e g and he references heren Unlke he case of marces over he feld of complex numbers localzaon heorems for uaernonc marces have been proposed for lef and rgh egenvalues separaely n Osrowsk and Brauer ype heorems for he rgh egenvalues of a uaernonc marx wh all real dagonal enres have been nroduced n 39 A Brauer ype heorem for he lef egenvalues of a uaernonc marx has been consdered n 6 Theorem 4 for he deleed absolue row sums whch s no same for he deleed absolue column sums of a uaernonc marx Smlar dfferences arse on he Gerschgorn and Osrowsk ype heorems for a uaernonc marx Therefore more research s reured o undersand he Osrowsk Gershgorn and Brauer ype heorems for marces over a uaernon dvson algebra Furhermore o nvesgang her applcaons n fndng varous bounds for he zeros of uaernonc polynomals and o 2 Mahemacs Subjec lassfcaon 2E5; 34L5; 5A8; 5A66 Keywords Skew feld; uaernonc marx; lef and rgh egenvalues; Gerschgorn ype heorems; Brauer ype heorem; uaernonc polynomals; uaernonc companon marces; sable uaernonc marx Receved: 27 January 27; Acceped: 9 Ocober 27 ommuncaed by Dragana vekovć Ilć orrespondng auhor: Sk Safue Ahmad The second auhor was funded by he SIR Gov of Inda Emal addresses: safue@acn safue@gmalcom Sk Safue Ahmad) skhara@acn Iskhar Al)

2 S S Ahmad I Al / Floma 32:2 28) analyze condons for he sably of a uaernonc marx one has o do furher research n hs drecon Therefore we have developed a general framework usng generalzed Holder neualy of uaernons o enhance our heory In he frs par of hs paper we provde a general framework for localzaon heorems for uaernonc marces Le M n H) be he space of all n n uaernonc marces Then for any A a j ) M n H) we prove a Osrowsk ype heorem whch saes ha all he lef egenvalues of A are locaed n he unon of n balls T A) : {z H : z a r A) γ c A) γ } where r A) : n j j a j and c A) : n j j a j γ From hs resul we deduce a suffcen condon for nverbly of a uaernonc marx We fnd ha he Brauer ype heorem proved n 6 Theorem 5 for he lef egenvalues n he case of deleed absolue column sums of a uaernonc marx s ncorrec and we prove a correced verson In fac n he case of he generalzed Hölder neualy over he skew feld of uaernons we show ha all he lef egenvalues of A a j ) M n H) are conaned n he unon of n generalzed balls: B A) : {z H : z a n ) γ r A) γ n p) A)) γ } where γ n p) A) : n j j a j p) p for any p ) wh p + Furher we prove ha all he rgh egenvalues of A M nh) wh all real dagonal enres are conaned n he unon of n generalzed balls B A) In he seuel we presen localzaon heorems for he rgh egenvalues of uaernonc marces In he second par of hs paper we provde bounds for he zeros of uaernonc polynomals usng he aforemenoned localzaon heorems Recall ha uaernonc polynomals n general are expressed n he followng forms z) : m z m + m z m + + z + ) p r z) : z m m + z m m + + z + 2) where j z H j m) The polynomals ) and 2) are called smple and monc f m Some recen developmens on he locaon and compuaon of zeros of uaernonc polynomals can be found n As a conseuence of he localzaon heorems for uaernonc marces we provde sharper bounds compared o he bound nroduced by G Opfer n 24 for he zeros of uaernonc polynomals Fnally we provde bounds for he zeros of uaernonc polynomals n erms of powers of he companon marces assocaed wh he uaernonc polynomals ) and 2) Some of our bounds are sharper han he bound from 24 The paper s organzed as follows: Secon 2 revews some exsng resuls from Secon 3 dscusses he Greshgorn ype Osrowsk ype and Brauer ype heorems for he lef and rgh egenvalues of a uaernonc marx Secon 4 explans bounds for he zeros of z) and p r z) omparsons are made wh he bound provded n 24 A suffcen condon for he sably of a uaernonc marx s also gven Secon 5 nroduces bounds for he zeros of he polynomals z) and p r z) n erms of powers of her companon marces Fnally Secon 6 summarzes hs work 2 Prelmnares Noaon: Throughou he paper R and denoe he felds of real and complex numbers respecvely The se of real uaernons s defned by H : { a + a + a 2 j + a 3 k : a a a 2 a 3 R } wh 2 j 2 k 2 jk The conjugae of H s : a a a 2 j a 3 k and he modulus of s : a 2 + a2 + a2 2 + a2 Ia) denoes he magnary par of a The real par of a uaernon 3 a + a + a 2 j + a 3 k s defned as R) a The collecon of all n-column vecors wh elemens n H s denoed by H n For x K n where K {R H} he ranspose of x s x T If x x x n T he conjugae of x s defned as x x x n T and he conjugae ranspose of x s defned as x H x x n For x y H n he nner produc s defned as x y : y H x and he norm of x s defned as x : x x The ses of m n real complex and uaernonc marces are denoed by M m n R) M m n ) and M m n H) respecvely

3 S S Ahmad I Al / Floma 32:2 28) When m n hese ses are denoed by M n K) K {R H} For A M m n K) he conjugae ranspose and conjugae ranspose of A are defned as A a j ) A T a j ) M n m H) and A H A) T M n m H) respecvely For z H n he vecor p-norm on H n s defned by z p : n z p ) /p where p < and z : { z } Defne R + : {α : α R α > } The se n : {r H : r ρ ρ f or all ρ H} s called an euvalence class of H Le x H n Then x can be unuely expressed as x x + x 2 j where x x 2 n Defne he funcon ψ : H n 2n by ψ x : x x 2 Ths funcon ψ s an njecve lnear ransformaon from H n o 2n Defnon 2 Le A M n H) Then A can be unuely expressed as A A + A 2 j where A A 2 M n ) Defne he funcon Ψ : M n H) M 2n ) by A A 2 Ψ A : A 2 A The marx Ψ A s called he complex adjon marx of A Defnon 22 Le A M n H) Then he lef rgh and he sandard egenvalues respecvely are gven by Λ l A) : { λ H : Ax λx for some nonzero x H n} Λ r A) : { λ H : Ax xλ for some nonzero x H n} and Λ s A) : { λ : Ax xλ for some nonzero x H n Iλ) } Defnon 23 Le A M n H) Then he marx A s sad o be sable f and only f Λ r A) H : { H : R) < } Defnon 24 Le A M n H) Then A s sad o be η-herman f A A η ) H where A η η H Aη and η { j k} Defnon 25 A marx A M n H) s sad o be nverble f here exss B M n H) such ha AB BA I n where I n s he n n deny marx We nex recall he followng resul necessary for he developmen of our heory Theorem Theorem 43 Le A M n H) Then he followng saemens are euvalen: a) A s nverble b) Ax has he unue soluon c) deψ A ) d) Ψ A s nverble e) A has no zero egenvalue Le A : a j ) M n H) and defne he absolue row and column sums of A as r A) : r A) + a and c A) : c A) + a n) 3 Dsrbuon of he lef and rgh egenvalues of uaernonc marces I s known from 29 orollary 32 ha a uaernonc marx A and s conjugae ranspose A H have he same rgh egenvalues However A and A H may no have he same lef egenvalues ake for example A and A j H We now presen he followng lemma for lef egenvalues of A and A j H Lemma 3 Le A M n H) and le λ H Then λ s a lef egenvalue of A f and only f λ s a lef egenvalue of A H

4 S S Ahmad I Al / Floma 32:2 28) Proof Le λ be a lef egenvalue of A Then here exss x ) H n such ha A λi n )x Ths can be wren as Ψ A λin )ψ x Hence follows ha λ s a lef egenvalue of A f and only f de Ψ A λin ) de Ψ H A λi n ) de ΨA λin ) H de ΨA H λi n ) Thus λ s a lef egenvalue of A H The Gerschgorn ype heorem s proved n 38 for he lef egenvalues usng deleed absolue row sums of a marx A M n H) However he Gerschgorn ype heorem for he lef egenvalues usng deleed absolue column sums of A has no ye been esablshed We now sae and prove he heorem Theorem 32 Le A : a j ) M n H) Then all he lef egenvalues of A are locaed n he unon of n Gerschgorn balls Ω A) : {z H : z a c A)} n ha s Λ l A) ΩA) : n Ω A) Proof Le λ be a lef egenvalue of A Then from Lemma 3 λ s a lef egenvalue of A H Then here exss some nonzero x H n such ha A H x λx Le x : x x n T H n and le x be an elemen of x such ha x x n Then x > From he -h euaon of A H x λx we have a j x j λx j Ths shows λ a a j : c A) j j We now have he followng localzaon heorem for he deleed absolue row and column sums of a marx A M n H) whch s known as Osrowsk ype heorem Theorem 33 Osrowsk ype heorem for he lef egenvalues) Le A : a j ) M n H) and le γ Then all he lef egenvalues of A are locaed n he unon of n balls T A) : {z H : z a r A) γ c A) γ } n ha s Λ l A) TA) : n T A) Proof Le γ ); as he cases γ and γ Gerschgorn ype heorems for column and row sums respecvely) can be obaned by akng lms We may assume ha all r A) > because we may perurb A by nserng a small nonzero enry ɛ > no any row n whch r A) ; he resulng marx has a ball ha s larger he ball for A and he resul follows n he lm as he perurbaon goes o zero Le λ be a lef egenvalue of A Then here exss some nonzero x H n such ha Ax λx Le x x x n T H n Then for each 2 n we have λ a x jj a j x j jj a j x j jj a j γ a j γ x j ) Applyng he generalzed Holder neualy wh p γ and γ we oban λ a x γ a j γ ) /γ jj jj a j γ jj a j γ x j ) γ jj a j x j γ γ γ r A) γ jj a j x j γ γ 3)

5 Snce r A) > hen from 3) we have Summng over all one obans ) λ γ a r A) γ x γ If S S Ahmad I Al / Floma 32:2 28) jj ) λ γ a r A) γ x γ a j x j jj γ a j x j γ c j A) x j γ 4) j ) λ γ a r A) γ > c A) for each such ha x hen 4) could no hold Hence we can conclude ha a leas one exss such as ) λ γ a r A) γ c A) ha s λ a r A) γ c A) γ Thus all he lef egenvalues of A are locaed n he unon of n balls T A) orollary 34 For any A : a j ) M n H) n 2 and for any γ Le us assume ha a > r A) γ r A) γ n 5) Then A s nverble Proof On he conrary suppose A s no nverble Then by Theorem 26 here s a lef egenvalue λ of A Now from Theorem 33 we oban a r A) γ c A) γ Ths conradcs our assumpon 5) Hence A s nverble The Brauer ype heorem s proved n 6 for he lef egenvalues n he case of deleed absolue column sums of a marx A M n H) Tha s f λ Λ l A) hen s conjugae λ les n he unon of nn ) 2 ovals of assn However hs s ncorrec as he followng example suggess: k Example 35 Le A Then by 6 Theorem 5 oval of assn s gven by { z H : z z j } j Here s a lef egenvalue of A and s conjugae s no conaned n he above oval of assn Accordng o 6 Theorem 5 f λ Λ l A) hen λ n j F j A) where j F j A) : { z H : z a z a jj c A)c j A) } j n j However hs resul s no necessarly rue as λ a λ a jj > c A)c j A) j n j whch follows from Example 35 Now we derve a correced verson of 6 Theorem 5 as follows: Theorem 36 Le A : a j ) M n H) Then all he lef egenvalues of A are locaed n he unon of nn ) 2 ovals of assn F j A) : { z H : z a z a jj c A)c j A) } j n j ha s Λ l A) FA) : n j F j A) j

6 S S Ahmad I Al / Floma 32:2 28) Proof Le λ be a lef egenvalue of A Then by Lemma 3 λ s a lef egenvalue of A H so ha here exss some nonzero x H n such ha A H x λx Le x : x x n T H n and le x s be an elemen of x such ha x s x n Then x s > learly f all he oher elemens of x are zero hen he reured resul holds Le x s and x be wo nonzero elemens of x such ha x s x x n s From he s-h euaon of A H x λx we have n j a jsx j λx s whch mples λ a ss )x s n j j s a jsx j Thus ) x λ a ss c s A) 6) x s Smlarly from A H x λx we oban ) xs λ a c A) x ombnng 6) and 7) we have λ a ss λ a c s A)c A) Hence all he lef egenvalues of A are locaed n he unon of nn ) 2 ovals of assn F j A) j n j For A : a j ) M n H) defne n p) A) : j j p a j p n p ) We are now ready o derve he followng localzaon heorem for lef egenvalues of a uaernonc marx Theorem 37 Le A : a j ) M n H) and le γ Then all he lef egenvalues of A are conaned n he unon of n generalzed balls } B A) : {z H : z a n ) γ r A) γ n p) A)) γ n 7) ha s for any p ) wh p + Λ l A) BA) : n B A) Proof Le µ be a lef egenvalue of A Then here exss some nonzero x H n such ha Ax µx Le x : x x n T H n and le x be an elemen of x such ha x x n Then from Ax µx we have a x + a j x j µx j j Ths mples µ a x j j a j x j j j Applyng he generalzed Hölder neualy o 8) we have µ a x a j p p x j j j a j x j 8) j j

7 S S Ahmad I Al / Floma 32:2 28) Snce x x for all n we have µ a x n p) A) n ) x ) ha s µ a n p) A) n ) Smlarly usng x x n) n 8) we ge µ a j j a j r A) 9) ) ombnng 9) and ) for γ we have ha s µ a γ n p) A)) γ n ) γ and µ a γ r A) γ ) µ a n ) γ Le us relae Theorem 37 o some exsng resuls: n p) A)) γ r A) γ Seng p 2 and γ mples ha he lef egenvalues of A : a j ) M n H) are conaned n he unon of n Greschgorn balls B A) : {z H : z a r A)} n ha s Ths resul can be found n 38 Theorem 6 Λ l A) BA) : n B A) Seng p 2 and γ mples ha he lef egenvalues of A : a j ) M n H) are conaned n he unon of n balls B A) : { z H : z a n ) 2 n 2) A) } n ha s Ths resul can be found n 36 Theorem Λ l A) BA) : n B A) We now presen a generalzaon of 38 Theorem 7 and 39 Theorem 3 by applyng he generalzed Hölder neualy over he skew feld of uaernons For a general marx A : a j ) M n H) all he rgh egenvalues may no le n he unon of n generalzed balls B A) n On he oher hand we show ha every conneced regon of he generalzed balls B A) n conans some rgh egenvalues of A Theorem 38 Le A : a j ) M n H) and le γ For every rgh egenvalue µ of A here exss a nonzero uaernon β such ha β µβ whch s also a rgh egenvalue) s conaned n he unon of n generalzed balls } B A) : {z H : z a n ) γ r A) γ n p) A)) γ n ha s { z µz : z H } n B A) where p ) wh p + Proof Le µ be a rgh egenvalue of A Then here exss some nonzero vecor x H n such ha Ax xµ Le x : x x n T H n and choose x from x as gven n Theorem 37 onsder ρ H such ha x µ ρx Then we have ρ a x a j x j a j x j 2) j j j j Usng he mehod from he proof of Theorem 37 we have ρ a n ) γ n p) A)) γ r A) γ Le us relae Theorem 38 o some exsng resuls:

8 Subsung p 2 and γ we oban Ths resul can be found n 38 Theorem 7 Subsung p 2 and γ we ge {z µz : z H} n Ths resul can be found n 39 Theorem 3 S S Ahmad I Al / Floma 32:2 28) {z µz : z H} n {z H : z a r A)} { z H : z a n n 2) A) } We nex presen a suffcen condon for he sably of a marx A M n H) Proposon 39 Le A : a j ) M n H) and le γ Assume ha Ra ) + n ) γ r A) γ n p) A)) γ < n 3) where p + wh p ) Then he marx A s sable Proof Le λ Λ r A) From Theorem 38 here exss ρ H such ha ρ λρ n B A) Whou loss of generaly we assume ρ λρ B l A) ha s ρ λρ a ll n ) γ r l A) γ n p) A)) γ l onsder λ : λ + λ 2 + λ 3 j + λ 4 k and a ll a l + b l + c l j + d l k Then from 3) we oban λ a l ) + ρ λ 2 ρ b l ) + ρ λ 3 jρ c l j) + ρ λ 4 kρ d l k) < Ra ll ) a l 4) The eualy 4) s possble when λ < ha s Rλ) < hence λ H Ths shows ha he marx A s sable When all he dagonal enres of a marx A M n H) are real we have he followng heorem Theorem 3 Le A : a j ) M n H) wh a R and le γ Then all he rgh egenvalues of A are conaned n he unon of n generalzed balls B A) : {z H : z a n ) γ r A) γ n p) A)) γ } n ha s Λ r A) BA) : n B A) where p ) wh p + Proof Le λ be a rgh egenvalue of A Then here exss some nonzero vecor x H n such ha Ax xλ Le x : x x n T H n and le x be an elemen of x such ha x x n Then x > Thus from Ax xλ we have a x + a j x j x λ j j snce a R so a x x a Then from he proof mehod of Theorem 37 we have λ a n ) γ n p) A)) γ r A) γ The above resul has grea sgnfcance as Herman and η-herman marces have all real dagonal enres In general η-herman marces arse wdely n applcaons To ha end we sae he followng proposon when all dagonal enres of A M n H) are real In parcular hs resul gves a suffcen condon for he sably of a marx A M n H)

9 S S Ahmad I Al / Floma 32:2 28) Proposon 3 Le A : a j ) M n H) wh a R and le γ Assume ha a + n ) γ r A) γ n p) A)) γ < n where p ) wh p + Then he marx A s sable From Theorem 3 all he complex rgh egenvalues of a marx A a j ) M n H) wh all real dagonal enres le n he unon of n-dscs D A) : {z : z a n ) γ r A) γ n p) A)) γ } n ha s Λ c A) DA) : n D A) However f dagonal enres are from \ R hen s no necessary ha all he complex rgh egenvalues of A are conaned n he unon of n-dscs D A) n as he followng examples sugges 2 j k Example 32 Le A : 2 The se of complex rgh egenvalues of A s k 3 + Λ c A) : {λ λ 2 λ 3 λ 4 λ 5 λ 6 } where λ λ λ λ 4 2 λ and λ For γ n 5) he dscs D A) D 2 A) and D 3 A) are as follows: D A) : {z : z + 2 2} D 2 A) : {z : z + 2 } and D 3 A) : {z : z 3 } From Fgure s clear ha λ λ 3 and λ 6 le ousde he dscs D A) D 2 A) and D 3 A) 5) 3 2 λ λ 3 D 3A) λ 5 Y axs D A) D 2A) λ 6 2 λ 2 λ Λ c A) X axs Fgure : Locaon of he complex rgh egenvalues of A from Example j + 2k j Example 33 Le A + j 2j k 2j + 2k In hs example here are sx complex rgh 3 + 2j 3k 8 egenvalues λ j j 6) whch are shown n Fgure 2 Subsung γ n 5) hen all he complex rgh egenvalues of he marx A are conaned n he unon of hree dscs D A) D 2 A) and D 3 A) where D A) : {z : z + 4 3} D 2 A) : {z : z } and D 3 A) : {z : z + 8 7}

10 S S Ahmad I Al / Floma 32:2 28) From Fgure 2 he sandard rgh egenvalues of A are λ λ 3 and λ 5 Then Λ r A) λ λ 3 λ 5 Also from Fgure 2 we observe ha Rλ ) H 3 5) Hence Rλ ) Rρ λ ρ) Rλ 2 ) Rτ λ 2 τ) and Rλ 3 ) Rν λ 3 ν) ρ τ ν H Thus he marx A s sable Y axs D 3 A) λ λ 2 D 2 A) λ 3 D A) λ 5 λ 6 λ Λ c A) X axs Fgure 2: Locaon of he complex rgh egenvalues of A from Example 33 In general smlar uaernonc marces may no have he same lef egenvalues see 38 Example 33 However he followng resul s rue Proposon 34 Le A M n H) and le W be any nverble real marx Then A and WAW have he same lef egenvalues Proof Le λ be a lef egenvalue of A Then here exss some nonzero vecor x H n such ha Ax λx Le W be an nverble real marx Then WAx Wλx λwx Now WAW Wx λwx Seng Wx y mples WAW y λy Le A : a j ) M n H) Suppose W dagw w 2 w n ) wh w R + n Then aj w ) W j AW and Λ l A) Λ l W AW) w Defne r W A) : j j a j w j w and c W A) : j j a j w w j n Applyng Theorem 33 o W AW we ge he followng heorem whch may be sharper han Theorem 33 dependng upon he choce of W Theorem 35 Le A : a j ) M n H) Then all he lef egenvalues of A are conaned n he unon of n balls T W A) : {z H : z a r W A)) γ c W A)) γ } n ha s Λ l A) Λ l W AW) T W A) : n TW A)

11 S S Ahmad I Al / Floma 32:2 28) Snce he above heorem holds for every W dagw w 2 w n ) where w R + we have Λ l A) Λ l W AW) W M n S) TW A) : T S A) where M n S) s a se of real dagonal marces wh non-negave enres Osrowsk ype se for he marx A Subsung γ n Theorem 35 we oban T S A) s called he mnmal Λ l A) Λ l W AW) η W A) : n ηw A) 6) where η W A) : { z H : z a r W A) } Therefore Λ l A) Λ l W AW) W M n S) ηw A) : η S A) where η S A) s called he frs mnmal Gerschgorn ype se for he marx A For γ n Theorem 35 we have Λ l A) Λ l W AW) Ω W A) : n ΩW A) 7) where Ω W A) : { z H : z a c W A) } Then Λ l A) Λ l W AW) W M n S) ΩW A) : Ω S A) where Ω S A) s called he second mnmal Gerschgorn ype se for he marx A 4 Bounds for he zeros of uaernonc polynomals In hs secon we derve bounds for he zeros of uaernonc polynomals by applyng he localzaon heorems for he lef egenvalues of a uaernonc marx Due o noncommuvy of uaernons we frs defne some basc facs on mulplcaon of uaernons For p H defne p : p For p H and H defne p : p : p p : p : p Recall he uaernonc polynomals z) and p r z) from ) and 2) Then he correspondng companon marces of he smple monc polynomals z) and p r z) are gven by pl : : m m m I pl m ) pl m 2 : m) and pr : T respecvely Le and defne smple monc reversal polynomals of z) and p r z) as follows: l z) : ) z m z m + z z m + + m z + r z) : z m p r z ) z m + z m + + z m + respecvely The correspondng companon marces of he smple monc reversal polynomals l z) and r z) are denoed by l and r respecvely We observe ha he zeros of l z) and r z) are he recprocal of zeros of z) and p r z) respecvely Now we need he followng resul:

12 S S Ahmad I Al / Floma 32:2 28) Proposon 4 32 Proposon Le λ H Then λ s a zero of he smple monc polynomal z) f and only f λ s a lef egenvalue of s correspondng companon marx pl In general a rgh egenvalue of pl s no necessarly a zero of he smple monc polynomal z) For example le a smple monc polynomal z) z 2 + jz + 2 Then s companon marx s gven by pl 2 j Here s a rgh egenvalue of pl However s no a zero of z) Analogous o Proposon 4 he followng resul s presened for p r z) Proposon 42 Le λ H Then λ s a zero of he smple monc polynomal p r z) f and only f λ s a lef egenvalue of s correspondng companon marx pr We now presen bounds for he zeros of z) as follows Theorem 43 Le z) be a smple monc polynomal over H of degree m Then every zero z of z) sasfes he followng neualy: r l ) γ c l ) γ)) z r ) γ c ) γ) m for every γ m Proof From Proposon 4 zeros of z) and lef egenvalues of pl are same Thus f z s a zero of z) hen z s a lef egenvalue of pl By applyng Theorem 33 Osrowsk ype heorem) o pl we oban z r ) γ c ) γ) m We use he respecve upper bounds for he zeros of he smple monc reversal polynomal l z) for he desred lower bounds for he zeros of z) orollary 44 Le z) be a smple monc polynomal over H of degree m Then every zero z of z) sasfes he followng neuales: { { } z + } + m ) 2 m ) { + m { } z m } Proof Subsung γ n Theorem 43 we oban he desred resuls Nex we derve he followng lemma whch gves a beer bound han Opfer s bound 24 Theorem 42 for { Lemma 45 Assume ha Then α T where α : + } and T : { m } m Proof ase : If hen α { + m } + m ase 2: If > hen α m ) { + } { + } and T : { m } m m { + } or m ) { + } and T : { m } { + m } + m Thus α T Ths complees he proof On he oher hand f < n Lemma 45 hen α T or α > T For example for a smple monc polynomal p l z) : z j + 2k)z 2 2kz + 5k we have α 4 and T 55 Hence α < T Furher f we consder p l z) z3 + 5jz j)z + 5 hen α 5 and T 36 Hence α > T Nex by applyng Theorem 33 o W pl W and W l W W s an nverble real dagonal marx) we oban dfferen and poenally sharper bounds

13 S S Ahmad I Al / Floma 32:2 28) Theorem 46 Le w R + m Then every zero z of he smple monc polynomal z) sasfes he followng neualy: m { r W l W ) γ c W l W ) γ} { z r W W ) γ c W W ) γ} m where W : dagw w 2 w m ) and γ Proof The companon marx of z) s gven by Then m m I pl m m W pl W m dag w w 2 w m w m w w m w 2 m By Proposon 34 pl and W pl W have he same lef egenvalues Res of he proof follows from he proof mehod of Theorem 43 orollary 47 Le z) be a smple monc polynomal over H of degree m Then every zero z of z) sasfes he followng neuales: { } w j + w m m j ) { } wj + w m j z where w j m d j+ j m w j+ m 2 w j w m w j m w m j m w j+ w + z j m w j+ w + Proof Subsung γ n Theorem 46 we ge he desred resuls Le w j w m j j m n he par ) of orollary 47 Then we oban { z 2 } j j m Ths s called he Kojma ype bound for he zeros of he smple monc polynomal z) For compuaon of bounds of he zeros of p r z) we defne he followng polynomal: z) : p r z) : j+ m j z j j H Now we dscuss he followng heorem whch shows relaon beween he zeros of p r z) and z) Theorem 48 Le λ H Then λ s a zero of he smple monc polynomal p r z) f and only f λ s a zero of he smple monc polynomal z) j w m ) Proof The correspondng companon marces of p r z) and z) are gven by pr : T and pl : H p r respecvely By Lemma 3 f λ s a lef egenvalue of pr hen λ s a lef egenvalue of H p r pl By Proposons 4 and 42 he lef egenvalues of pr and pl mply he zeros of p r z) and z) respecvely Hence f λ s a zero of p r z) hen λ s also a zero of z) Remark 49 Smlar resuls can be obaned for he uaernonc polynomal p r z) as well

14 S S Ahmad I Al / Floma 32:2 28) Bounds for he zeros of uaernonc polynomals by usng he powers of companon marces We presen some prelmnares resuls for he powers of companon marces pl and pr In general f λ s a lef egenvalue of a uaernonc marx A hen λ 2 s no necessarly a lef egenvalue of A 2 For example for a uaernonc marx A we have Λ l A) : { µ : µ α + βj + γk α 2 + β 2 + γ 2 } and A 2 So Λ l A 2 ) : {} Here j s a lef egenvalue of A bu j 2 s no a lef egenvalue of A 2 Proposon 5 If λ s a lef egenvalue of pl wh respec o he egenvecor x H n hen λ s a lef egenvalue of correspondng o he same egenvecor x H n Proof ase a): Le be a posve neger and le λ be a lef egenvalue of pl Then here exss x : λ λ 2 λ m T H n such ha pl x λx Therefore 2 x pl pl x) pl xλ xλ 2 x pl x) xλ xλ λ x Thus λ s a lef egenvalue of marx correspondng o he same egenvecor x H n ase b): Le be a negave neger From ase a) we have pl x xλ Ths mples x xλ Therefore 2 x x +) x) xλ xλ 2 x) +) xλ xλ λ x Thus λ s a lef egenvalue of wh respec o he same egenvecor x H n Nex we sae he followng resul for lef egenvalues of pr and p r s a nonzero neger) Proposon 52 If λ s a lef egenvalue of pr wh respec o he egenvecor x H n hen λ s a nonzero neger) s a lef egenvalue of p r correspondng o he same egenvecor x H n Proof ase a): Le be a posve neger and le λ be a lef egenvalue of pr Now from Lemma 3 λ s a lef egenvalue of H p r Then here exss x : λ λ) 2 λ) m H n such ha H p r x λx xλ Ths gves H pr ) 2 x H pr H p r x) H p r xλ xλ) 2 ) H pr x ) H pr H pr x) ) H p r xλ xλ) λ) x Thus λ) s a lef egenvalue of ) H p r Then by Lemma 3 λ s a lef egenvalue of p r ase b): Le be a negave neger From ase a) we have H p r x λx xλ Ths mples H p r ) x xλ) Thus H p r ) 2 x H p r ) { H p r ) x} H p r ) xλ) xλ) 2 H p r ) x H p r ) +) { H p r ) x} H p r ) +) xλ) xλ) λ) x

15 S S Ahmad I Al / Floma 32:2 28) Thus λ) s a lef egenvalue of H p r ) Then by Lemma 3 λ s a lef egenvalue of p r Furher we presen a framework o fnd he powers of he companon marx pl whch can be derved n a smple procedure as follows keepng n vew ha uaernons do no commue m m I Theorem 53 onsder pl pl m ) pl m 2 : m) a) If < m s a posve neger hen m m I D 8) b) f m hen where m ) m : m) m 2) m : m) m : m) m : m) m m m ) : m m) pl m ) m 2 : m) : 9) m : m ) + m m) pl m 2 : m) pl m : ) 2 m : ) : m : ) Noe ha pl k : m) denoes he k-h row of he marx pl pl m + : m) 2 m + : m) and D : m + : m) m m ) m I Proof Assumng 8) becomes pl where pl m ) pl m 2 : m) pl m ) : pl m 2 : m) : m Thus he heorem s rue for Now le us consder pl as m k k k A pl B m k D where A : pl : k : m k) B : pl k + : m m k + : m) : pl k + : m : m k) D : pl k + : m m k + : m) For k 3 we ge 3 2 m 2 m 2 I 2 D m A B m 2 D m 2 2 m 2 D 2 A + D B + DD Noe ha n each sep sze of he deny marx I reduces by order and he sze of marx ncreases by order Smlarly he marx D ncreases by row and decreases by column Fnally afer rearrangng

16 and separang and I marces we ge S S Ahmad I Al / Floma 32:2 28) m 2 m 2 I 2+ D where and D are of sze 3 3 and 3 m 3) respecvely Assumng ha he heorem s rue for k we have k+ k pl m k m k D k A + D B + DD k k+ m k m k I k+ D where he correspondng and D marces are gven n he saemen of he heorem The proof for m s smlar In he case of uaernonc marx pl T p r bu p r ) T for 2 Ths s llusraed by he followng example Example 54 onsder he followng smple monc polynomals over H : z) z 3 kz 2 + k j)z + + j) and p r z) z 3 z 2 k + zk j) + + j) The correspondng companon marces of z) and p r z) are gven by 2 2 I pl pl 3 ) pl 3 2 : 3) and pr T respecvely where pl 3 ) j and pl 3 2 : 3) : j k k Then j j 2 j j k k and 2 p r j k j j 2 j j k k j k Ths shows ha 2 p r 2 ) T Hence we can derve resuls analogous o Theorem 53 for he case of p r 2 m Theorem 55 onsder pr pr m) m I pr 2 : m m) a) If < m s a posve neger hen m p r m I D 2) b) f m hen where p r m ) p r : m m) m 2) p r : m m) p r : m m) p r : m m) m m : pr : m) 2 p r : m) p r : m) D : pr + : m m) 2 p r + : m m) p r + : m m) p r m) : pr m) p r m m) and p r 2 : m m) : p r : m m) + pr 2 : m m) m m) p r

17 S S Ahmad I Al / Floma 32:2 28) Proof The proof follows from he proof mehod of Theorem 53 Polynomals from Example 54 sasfy z) : p r z) z 3 + kz 2 + j k)z + j) and p r z) : z) z 3 + z 2 k + zj k) + j) Thus he companon marces correspondng o z) and p r z) are gven by pl pl and pr pr respecvely Nex + j j 2 + j j + k k and 2 p r j + k j j + j k j k j + k Then Now we have r 3 r )) / and 3 r )) /2 2 pr 2 )) /2 3 r 2 p r )) / and r 3 2 )) / r 2 p r )) /2 and 3 r )) /2 2 pr Furher we have he followng bounds for he zeros of z) and p r z) for γ r 3 2 )) /2 Theorem 56 Le z) and p r z) be he smple monc polynomals over H of degree m and le and p r 2) be he -h power of he companon marces pl and pr correspondng o z) and p r z) respecvely Then for γ bounds for every zero z of z) sasfy he followng neuales: m m r l )) γ/ c l )) γ)/ ) z m r )) γ/ )) γ)/ ) r c r z m r and bounds for every zero z of p r z) sasfy he followng neuales: m m r r r )) γ/ c l )) γ/ c r )) γ)/ ) z m l )) γ)/ ) z m )) γ/ )) γ)/ pl c pl 2) r )) γ/ )) γ)/ pr c pr 22) r r )) γ/ )) γ)/ pr c pr 23) )) γ/ )) γ)/ pl c 24) pl Proof Le λ be a lef egenvalue of pl Then by Proposon 5 λ 2 s posve neger) s a lef egenvalue of Hence by applyng Theorem 33 we ge 2) By Lemma 3 λ s a lef egenvalue of pr and by Proposon 52 λ) s a lef egenvalue of pr ) Then from Theorem 33 22) follows The proof of 23) and 24) are smlar Subsung 2 and γ n Theorem 56 we have he followng corollary orollary 57 Le z) and p r z) be he smple monc polynomals over H of degree m Then bounds for every zero z of z) sasfy he followng neuales: β z α and β 2 z α 2

18 S S Ahmad I Al / Floma 32:2 28) where /2 m /2 m α j m j j j j { α 2 + m ) /2 + m ) /2 + j + j m j ) } /2 2 j m m β β 2 2 j m j j /2 m m j j { + ) /2 + m j + m j m j+ ) /2 } m j+ /2 m + m ) /2 and bounds for every zero z of p r z) sasfy he followng neuales: where α 3 2 j m m α 4 j β 3 z α 3 and β 4 z α 4 { + m ) /2 + m ) /2 + j + j m j ) /2 } j /2 /2 m m j j j β 3 { + 2 j m ) /2 m + m ) /2 + m j + m j m j+ ) /2 m β 4 j j /2 m j m j m j+ Proof The proof follows from Theorem 56 and Appendx A Example 58 onsder he followng polynomals z) and p r z) over H: /2 m+ m z) z k)z j)z k)z j)z k)z + 6j 2) p r z) z 6 + z 5 + 3k) + z j) + z k) + z j) + z6 + 8k) + 6j 2) The zeros of z) are gven n 32 Moreover we fnd he zeros of p r z) by Nven s algorhm 23 Thus he zeros and bounds for he zeros of z) and p r z) are gven n he followng ables 6 oncluson In hs paper we have derved Osrowsk ype heorem for lef egenvalues of a uaernonc marx ha generalzes Osrowsk ype heorem for rgh egenvalues of a uaernonc marx when all he dagonal enres of a uaernonc marx are real We have derved a correced verson of he Brauer ype heorem for lef egenvalues for he deleed absolue column sums of a uaernonc marx Moreover we have exended localzaon heorems by applyng he generalzed Hölder neualy for lef as well as rgh egenvalues of a uaernonc marx Bounds for he zeros of uaernonc polynomals have derved As

19 S S Ahmad I Al / Floma 32:2 28) z z z 2 z 2 2k k k k Table : Zeros of z) and p r z) and her absolue values where z and z 2 are he se of zeros of z) and p r z) respecvely Example 58 lower bound upper bound orollary 44 ) orollary 44 2) Theorem 43 γ / Table 2: Lower and upper bounds for he zeros of z) and p r z) Example 54 lower bound lower bound orollary 57 a) orollary 57 b) orollary 57 2a) orollary 57 2b) Table 3: Lower and upper bounds for he zeros of z) and p r z) a conseuence we have shown ha some of our bounds are sharper han he bound gven n 24 Furher we have derved bounds va he powers of companon marces whch are always sharper han he bound gven n 24 Appendx A In hs appendx we sae formulas for he suares of uaernonc companon marces For 2 Theorem 53 mples 2 2 m 2 m 2 I 2 D where : pl m : 2) 2 m : 2) m m and pl m 3 : m) D m m 3 : m) m 2 m 3 m ) 2 m m 2 m 2 I 2 D where pl m : 2) 2 m : 2) m m and pl m 3 : m) D 2 m 3 : m) 2 3 m m 2 m 3 m ) 2 m 2 2 l 2 m 2 m 2 I 2 D where m m

20 S S Ahmad I Al / Floma 32:2 28) and and 2 l D m 2 m 2 m ) m 2 m 2 I 2 D where m m m 2 D m 2 m ) 2 2 For 2 Theorem 55 mples m p r where m 2 I D pr : 2 m) 2 p r : 2 m) m m and 2 p r m m 2 I D D pr 3 : m m) 2 p r 3 : m m) where m 2 2 m 3 3 m 2 m m m ) 2 m 2 and D 2 2 m 3 3 m 2 m ) 2 m m 2 m r where m 2 I D m m m 2 m 2 m and D )2 2 2 r m m m m 2 I D and D where m 2 m 2 m ) 2 2

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