Eigenvalue localization for complex matrices

Transcription

1 Electroic Joural of Liear Algebra Volume 7 Article Eigevalue localizatio for complex matrices Ibrahim Halil Gumus Adıyama Uiversity, igumus@adiyama.edu.tr Omar Hirzallah Hashemite Uiversity, o.hirzal@hu.edu.jo Fuad Kittaeh Jorda Uiversity, fkitt@ju.edu.jo Follow this ad additioal works at: Recommeded Citatio Gumus, Ibrahim Halil; Hirzallah, Omar; ad Kittaeh, Fuad. (014), "Eigevalue localizatio for complex matrices", Electroic Joural of Liear Algebra, Volume 7. DOI: This Article is brought to you for free ad ope access by Wyomig Scholars Repository. It has bee accepted for iclusio i Electroic Joural of Liear Algebra by a authorized editor of Wyomig Scholars Repository. For more iformatio, please cotact scholcom@uwyo.edu.

2 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society Volume 7, pp , December EIGENVALUE LOCALIZATION FOR COMPLEX MATRICES IBRAHIM HALIL GÜMÜŞ, OMAR HIRZALLAH, AND FUAD KITTANEH Abstract. Let A be a complex matrix with 3. It is show that at least of the eigevalues of A lie i the disk z tra 1 A tra A A AA spd (A), where A,trA, ad spd(a)deote the Frobeiusorm, the trace, ad the spreadofa, respectively. I particular, if A = [a ij ] is ormal, the at least of the eigevalues of A lie i the disk z tra 1 A tra 3 max i,,..., a ki + a kj + k=1 k i k=1 k j aii ajj. Moreover, the costat 3 ca be replaced by 4 if the matrix A is Hermitia. Key words. Eigevalue, Localizatio, Frobeius orm, Trace, Spread, Normal matrix. AMS subject classificatios. 15A18, 15A4. 1. Itroductio. Let M (C) be the set of all complex matrices. For a matrixa M (C), let λ j (A), j = 1,...,, be the eigevaluesofarepeated accordig to multiplicity, ad let the symbols A ad tra deote the Frobeius orm ad the trace of A, respectively. We have to keep i mid that the Frobeius orm is uitarily ivariat, that is, UAV = A for all uitary matrices U,V i M (C) ad tra = λ j (A). A estimate for the eigevalues of matrices [14] says that if A is a real Received by the editors o March 14, 014. Accepted for publicatio o December 6, 014. Hadlig Editor: Brya L. Shader. Departmet of Mathematics, Faculty of Arts ad Scieces, Adıyama Uiversity, Adıyama, Turkey (igumus@adiyama.edu.tr). Departmet of Mathematics, Hashemite Uiversity, Zarqa, Jorda (o.hirzal@hu.edu.jo). Departmet of Mathematics, The Uiversity of Jorda, Amma, Jorda (fkitt@ju.edu.jo). 89

3 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society Volume 7, pp , December symmetric matrix, the Eigevalue Localizatio A tra. (1.1) for l = 1,...,. Moreover, a geeralizatio of (1.1) for arbitrary matrices A M (C) has bee obtaied i [11]. Oe of the iterestig estimates that presets a refiemet of (1.1) for oormal matrices has bee established i [1]. This refiemet asserts that if A M (C), the ( 1 A A A AA 6 A for l = 1,...,. A improvemet of the boud for λ l (A) tra bee established i [13], which asserts that if A M (C), the ) tra 1 4 A tra A A AA for l = 1,...,. (1.) give i (1.) has (1.3) I this paper, we obtai bouds ad localizatio results for the eigevalues of matrices. Our results, which ivolve the traces ad the spreads of matrices, are better tha some kow bouds ad localizatio results. I particular, refiemets of (1.) ad (1.3) will be give.. Eigevalue localizatio for oormal matrices. I this sectio, we preset refiemets of (1.) ad (1.3) for oormal matrices. Throughout the paper, we let the symbol S l deote the set {1,...,}\{l} for l = 1,...,. We start with the followig result for scalars. Lemma.1. Let z 1,...,z be complex umbers such that for l = 1,...,. z l + 1 z j z k = 1 z j = 0. The z j

4 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society Volume 7, pp , December I.H. Gumus, O. Hirzallah, ad F. Kittaeh Proof. Let l {1,...,}. The z l + 1 z j z k = z j + 1 z j z k j S l = z j + 1 z j z k j S l = j j + 1 z j z k j Slz j Slz = z j + 1 ( z j z k + z k z j )+ 1 j S l z j z k = j S l z j + 1 = j S l z j + 1 = ( 1) j S l z j = ( 1) j k j k j k j k ( z j z k + z k z j + z j z k ) ( z j + z k ) z j ( 1) z l. (.1) It follows from the idetity (.1) that z l + 1 z j z k = ( 1) z j, ad so as required. z l + 1 z j z k = 1 z j, Aother result for scalars that we eed is the followig. Its proof is similar to that of Lemma.1 ad is left to the reader. Lemma.. Let z 1,...,z be complex umbers. The z j = 1 z j + 1 z j z k. 1 j<k

5 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society Volume 7, pp , December Eigevalue Localizatio 895 Our first result is the followig idetity. Theorem.3. Let A M (C) with 3. The + 1 λ j (A) λ k (A) = 1 λ j (A) tra for l = 1,...,. Proof. Let z j = λ j (A) tra, j = 1,...,. The = z l + 1 = 1 = 1 = 1 = 1 = 1 = λ j (A) λ k (A) z j z k z j (by Lemma.1) z j = 0, ad so 1 z j + 1 z j z k (by Lemma.) 1 j<k 1 z j z k 1 j<k 1 λ j (A) λ k (A) 1 j<k λ j (A) 1 λ j (A) (by Lemma.) λ j (A) tra for l = 1,...,. Also, we eed the followig boud for the eigevalues of a give matrix A M (C) [4].

6 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society Volume 7, pp , December I.H. Gumus, O. Hirzallah, ad F. Kittaeh Lemma.4. Let A M (C). The λ j (A) A tra A A AA + tra. Remark.5. The boud give i Lemma.4 is sharper tha the bouds: λ j (A) A A A AA 6 A ad λ j (A) A 4 A A AA give earlier i [] ad [7], respectively. These bouds, i tur, are sharper tha the classical Schur s iequality [15, p. 50]. Based o Theorem.3 ad Lemma.4, we have the followig localizatio result for the eigevalues of matrices. This result icludes a refiemet of (1.3). Theorem.6. Let A M (C) with 3. The 1 ( for l = 1,...,. I particular, + 1 A tra λ j (A) λ k (A) ) A A AA 1 4 A tra A A AA (.) (.3) for l = 1,...,. Proof. The result follows from Theorem.3 ad Lemma.4. We remark here that (.3) has bee obtaied earlier i [13]. Applicatios of Theorem.6 are give i the followig result. This result icludes a refiemet of (1.).

7 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society Volume 7, pp , December Corollary.7. Let A M (C) with 3. The: Eigevalue Localizatio 897 (a) 1 A 4 A A AA tra (.4) (b) for l = 1,...,. ( 1 A A A AA 4 A for l = 1,...,. ) tra (.5) Proof. By direct computatios, it ca be see that A tra A A AA A 4 A A AA tra (.6) ad A 4 A A AA A A A AA 4 A. (.7) Now, part (a) of the corollary follows from (.3) ad (.6), while part (b) follows from (.4) ad (.7). Remark.8. It is clear from (.6) ad (.7) that the boud for λ l (A) tra give i (.3) is sharper tha that give i (.4), which, i tur, is sharper tha that give i (.5). Aother boud for the eigevalues of a give matrix A M (C) says that λ j (A) A A A AA 4 A. This boud ca be iferred from Lemma.4, (.6), ad (.7), ad it ca also be obtaied from Theorem i [3], cocerig measures of oormality of matrices. To give aother applicatio of Theorem.6, we eed the followig lemma [9]. Lemma.9. Let z 1,...,z be complex umbers. The max z j z k z j z k. (.8) j,k=1,..., 1 j<k

8 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society Volume 7, pp , December I.H. Gumus, O. Hirzallah, ad F. Kittaeh Based o Theorem.6 ad Lemma.9, we have the followig result. This result presets aother refiemet of (1.3). Corollary.10. Let A M (C) with 3. The 1 A tra A A AA for l = 1,...,, where s(a) = mi 1 l max λ j (A) λ k (A). Proof. Sice S l cotais 1 umbers, we have λ j (A) λ k (A) = λ j (A) λ k (A) j<k Now, (.9) follows from (.) ad (.10). s (A) ( 1) max λ j (A) λ k (A) (by Lemma.9) (.9) ( 1)s (A). (.10) Remark.11. A localizatio result for the eigevalues of matrices has bee give i [4]. This result asserts that if the eigevalues of a matrix A M (C) are arraged as λ 1 (A) λ (A), the m m 1 A m tra A A AA + tra (tra) m for l = 1,...,m, where m is ay iteger satisfyig raka m. Aother result aalogous to (.) ca be stated as follows. Theorem.1. Let A M (C) with 3. The 1 ( + 1 A tra ( 1) ( tr ( A ) λ l(a) ) (tra λ l (A)) ) A A AA (.11)

9 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society Volume 7, pp , December Eigevalue Localizatio 899 for l = 1,...,. Proof. Observe that λ j (A) λ k (A) = (λ j (A) λ k (A)) (λ j (A) λ k (A)) = λ j (A)+λ ( k (A) λ j(a)λ k (A)) = λ j (A)+ λ k (A) λ j (A)λ k (A) = ( 1) λ j (A) j (A) j S j Slλ l = ( 1) λ j (A) λ l (A) λ k (A) λ l (A) = ( 1) λ j (A ) λ l (A) (tra λ l (A)) = ( 1) ( tr ( A ) λ l (A)) (tra λ l (A)). (.1) Now, the result follows from Theorem.6 ad (.1). The spreadspd(a) ofamatrix A M (C) is defied to be the maximum distace betwee ay two eigevalues of A, that is, spd(a) = max j,k=1,..., λ j(a) λ k (A). I the followig result, we describe a disk that cotais at least of the eigevalues of a give matrix A M (C). A localizatio result for at least of the eigevalues of A M (C) that improves (1.3) ca be cocluded from this result.

10 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society Volume 7, pp , December I.H. Gumus, O. Hirzallah, ad F. Kittaeh Theorem.13. Let A M (C) with 3. The at least of the eigevalues of A lie i the disk z tra 1 A tra A A AA spd (A). (.13) Proof. Let s,t {1,...,} be such that spd(a) = λ s (A) λ t (A) ad let S = {1,...,}\{s,t}. The s,t S l for all l S, ad so for all l S. It follows that λ j (A) λ k (A) = spd(a) = max λ j (A) λ k (A) (.14) j<k λ j (A) λ k (A) ( 1) max λ j (A) λ k (A) (by Lemma.9) = ( 1)spd (A) (by the idetity (.14)) (.15) for all l S. Now, (.15) ad Theorem.6 imply that + 1 spd (A) 1 A tra A A AA, (.16) ad so 1 A tra A A AA spd (A) (.17) for all l S. Sice the set S cotais umbers, the (.17) meas that at least of the eigevalues of A lie i the disk give i (.13). Remark.14. Theorem.13 guaratees that of the eigevalues lie i the disk (.13). The questio that arise here is what about the remaiig two eigevalues.

11 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society Volume 7, pp , December Eigevalue Localizatio 901 I fact, it is clear from the proof of Theorem.13 that the eigevalues of the matrix A M (C) where the spread is attaied do ot ecessarily lie i the disk (.13). Moreover, if oe of these eigevalues is ot simple, the accordig to the proof of Theorem.13, we ca see that this eigevalue must lie i this disk. The followig lemma eables us to give a ew boud for the eigevalues of matrices. The proof of this lemma follows by direct computatios. We leave the details for the iterested reader. Lemma.15. Let A M (C) with 3. The: (a) (b) λj (A) tra = λ j (A) tra. λ j (A) λ k (A) = ( ) l=1 1 j<k λ j (A) λ k (A). Theorem.16. Let A M (C) with 3. The λ j (A) ( 1) A tra A A AA + tra spd (A). Proof. Observe that λ j (A) tra λ j(a) tra = = λ j(a) tra l=1 λ j(a) tra l= spd (A) 1 j<k λ j (A) λ k (A) (by Lemmas.9 ad.15 (a)) λ j (A) λ k (A) l=1 (by Lemma.15 (b)) λ j (A) λ k (A)

12 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society Volume 7, pp , December I.H. Gumus, O. Hirzallah, ad F. Kittaeh l=1 1 ( A tra ) 1 A A AA (by Theorem.6) = ( 1) A tra A A AA. (.18) Now, the result follows from (.18). 3. Eigevalue localizatio for ormal matrices. I this sectio, we are iterested i estimates for at least of the eigevalues of a ormal matrix A M (C). I order to do this, we eed the followig two lemmas of Bhatia ad Sharma [1] ad Mirsky [9]. It should be metioed here that Bhatia ad Sharma did ot write the first lemma below explicitly i this form, but it ca be deduced from their results. Before presetig these lemmas, we eed to defie two fuctioals o M (C) as follows: Let ad dra = i, a ij where A = [a ij ] M (C). v(a) = dra A dra, Lemma 3.1. Let A = [a ij ] M (C) be ormal. The spd (A) max(α 1,β 1 ), (3.1) where ad α 1 = 3 β 1 = 3 max i,,..., a ki + k=1 k i a kj + a ii a jj k=1 k j ( ) A tra +v(a) RetrAdrA.

13 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society Volume 7, pp , December Eigevalue Localizatio 903 where Lemma 3.. Let A = [a ij ] M (C) be ormal. The spd(a) max(α,β ), (3.) ad β = max i,,..., α = 3 max i,,..., a ij a ii a jj + (a ii a jj ) +4a ij a ji + a ij + a ij. Theorem 3.3. Let A = [a ij ] M (C) be ormal with 3. The at least of the eigevalues of A lie i the disk z tra ( ) 1 A tra max(α 1,α,β 1,β ). (3.3) I particular, at least of the eigevalues of A lie i the disk z tra 1 A tra 3 max i,,..., a ij. (3.4) Proof. Let l S, where S is the set defied i the proof of Theorem.13. The + ( 1) max ( α 1,α,β 1,β ) + 1 spd (A) (by Lemmas 3.1 ad 3.) 1 A tra (by (.17)). (3.5) Now, (3.3) follows from Theorem.13 ad (3.5), while (3.4) is a special case of (3.3). The costat 3 i (3.4) ca be improved if the matrix A is Hermitia. This improvemet ca be achieved usig a result of Mirsky [10] that says if A = [a ij ] M (C) is Hermitia, the ( spd (A) max 4 a ij +(a ii a jj ) ). (3.6) i,,...,

14 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society Volume 7, pp , December I.H. Gumus, O. Hirzallah, ad F. Kittaeh I fact, through Lemma 3.1, Bhatia ad Sharma itroduced a remarkable improvemet of (3.6). This improvemet asserts if A = [a ij ] M (C) is Hermitia with 3, the spd (A) max a ik + a kj + (a ii a jj ). (3.7) i,,..., k=1 k i k=1 k j Based o (3.7) ad usig a proof similar to that give for Theorem 3.3, we have the followig result. Theorem 3.4. Let A = [a ij ] M (C) be Hermitia with 3. The at least of the eigevalues of A lie i the disk z tra 1 A tra max a ik + a kj (aii ajj) +. (3.8) i,,..., k=1 k i k=1 k j A applicatio of (3.4) ca be see as follows. Applicatios of (3.3) ad (3.8) ca be deduced by a similar argumet. Corollary 3.5. Let A = [a ij ] M (C) be ormal with. The all the eigevalues of A lie i the disk z tra 1 A tra 3 4 max i,,..., a ij. (3.9) [ ] A 0 Proof. Let B = [b ij ] =. The B M (C) is ormal ad the eigevalues of B are the same as those of A with duplicate multiplicities. It follows from 0 A (3.4), applied to the matrix B, that the disk z trb 1 B trb 3 max i,,..., b ij (3.10) cotais at least of the eigevalues of B. Sice the eigevalues of B are ot simple, it follows from Remark.14 that these eigevalues lie i the disk (3.10). I

15 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society Volume 7, pp , December Eigevalue Localizatio 905 particular, the eigevalues of A lie i this disk. Now, the result follows i view of the facts that B = A, trb = tra, ad max b ij = max a ij. i,,..., i,,..., Remark 3.6. Theorems3.3, 3.4, adcorollary3.5arebasedothelowerbouds for the spreads of ormal matrices metioed i this paper. Related localizatio results ca be obtaied usig further lower bouds for the spreads of ormal matrices (see, e.g., [5], [6], ad [8]). I the followig result, we utilize Theorem.6 ad the spectral theorem for ormal matrices to ivestigate the equality coditios of (1.1). Corollary 3.7. Let A M (C) with 3. The = 1 A tra. (3.11) for some l {1,...,} if ad oly if A is a scalar matrix or A is ormal with oly two distict eigevalues oe of multiplicity 1 ad the other is of multiplicity oe. Our fial result follows from Theorem.13, Corollary 3.7, ad Remark.14. Corollary 3.8. Let A M (C) with 3. If = 1 A tra (3.1) for some l {1,...,}, the A is ormal ad at least 1 of the eigevalues of A lie i the disk (.13). REFERENCES [1] R. Bhatia ad R. Sharma. Some iequalities for positive liear maps. Liear Algebra Appl., 436: , 01. [] P.J. Eberlei. O measures of o-ormality for matrices. Amer. Math. Moth., 7: , [3] L. Elser ad M.H.C. Paardekooper. O measures of oormality of matrices. Liear Algebra Appl., 9:107 13, [4] T.-Z Huag ad L. Wag. Improvig bouds for eigevalues complex matrices usig traces. Liear Algebra Appl., 46: , 007. [5] E. Jiag ad X. Zha. Lower bouds for the spread of a Hermitia matrix. Liear Algebra Appl., 56: , [6] C. Johso, R. Kumar, ad H. Wolkowicz. Lower bouds for the spread of a matrix. Liear Algebra Appl., 71: , [7] R. Kress, H.L. de Vries, ad R. Wegma. O oormal matrices. Liear Algebra Appl., 8:109 10, 1974.

16 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society Volume 7, pp , December I.H. Gumus, O. Hirzallah, ad F. Kittaeh [8] J. Merikoski ad R. Kumar. Characterizatios ad lower bouds for the spread of a ormal matrix. Liear Algebra Appl., 364:13 31, 003. [9] L. Mirsky. The spread of a matrix. Mathematika, 3:17 130, [10] L. Mirsky. Iequalities for ormal ad Hermitia matrices. Duke Math. J., 4: , [11] O. Rojo, H. Rojo, R. Soto, ad T.Y. Tam. Eigevalues localizatio for complex matrices. Proyeccioes, 11:11 19, 199. [1] O. Rojo, R.L. Soto, ad H. Rojo. New eigevalue estimates for complex matrices. Comput. Math. Appl., 5:91 97, [13] P. Sargolzaei ad R.R. Rakhshaipur. Improvig the localizatio of eigevalues for complex matrices. Appl. Math. Sci., 5: , 011. [14] P. Tarazaga. Eigevalue estimates for symmetric matrices, Liear Algebra Appl., 135: , [15] X. Zha. Matrix Iequalities. Lecture Notes i Mathematics, Vol. 1790, Spriger-Verlag, Berli, 00.