Improving the Localization of Eigenvalues for Complex Matrices

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1 Applied Mathematical Scieces, Vol. 5, 011, o. 8, Improvig the Localizatio of Eigevalues for Complex Matrices P. Sargolzaei 1, R. Rakhshaipur Departmet of Mathematics, Uiversity of Sista ad Baluchesta Zaheda, Ira Abstract I this paper, we give a ew boud for λ i, which improves Schur, Kress, ad Eberlei s iequalities. By usig this estimate, we provide disks ad rectagles which iclude the eigevalues of a complex matrix ad the we compare these regios. Fially, we improve these regios by a strip for matrices with characteristic polyomial icludig real coefficiets. Numerical examples are provided to illustrate the results i this cotributio. Mathematics Subject Classificatio: 65F15, 65B99 Keywords: Eigevalues, localizatio, complex matrices, rectagular regio 1 Itroductio Matrices are a key tool i liear algebra. Matrices fid may applicatios, for example, graph theory uses matrices to keep track of distaces betwee pairs of vertices i a graph. Computer graphics uses matrices to project - dimesioal space oto a -dimesioal scree. Matrix calculus geeralizes classical aalytical otios such as derivatives of fuctios or expoetials to matrices. Due to these, A major brach of umerical aalysis is devoted to the developmet of efficiet algorithms for matrix computatios, a subject that is ceturies old but still a active area of research. Let A be a complex matrix of order with eigevalues λ 1,λ,..., λ. Gersgori doe almost first step for localizatio of eigevalues. He showed each 1 Correspodig Author. sargolzaei@hamoo.usb.ac.ir

2 1858 P. Sargolzaei ad R. Rakhshaipur eigevalue λ of A satisfies at least i oe of the followig disks: λ a ii a ij,,,...,. j=1,j i Wolkowicz i [6, 7] provided upper ad lower bouds for (Rλ i ) ad (Iλ i ) ad the Rojo [, 4] provided regios that are cotai eigevalues. More precisely, Eberlei i [1] showed that λ i A AA A A = k 6 A 1 (A), (1) while Kress [] gave a tighter boud λ i ( A 4 AA A A ) 1/ = k (A). () Note that we use. as the Euclidea orm throughout. The rest of this paper is orgaized as comes ext. I Sectio, based o the available literature, we review a upper boud for λ i, which is tighter tha k (A). I sectio, we oce agai review the disks ad rectagles which cotais of eigevalues, accordig to the literature. The, we improve these regios by a strip for matrix with characteristic polyomial icludig real coefficiets. I fact, the cotributio of this paper will be furished i Sectio.1. The paper eds with several examples i Sectio 4. Prerequisites for λ i Theorem.1. Let A be a complex matrix of order with eigevalues λ 1,λ,..., λ. The λ i k (A), () where k (A) =(( A tra ) AA A A ) 1/ + tra. Proof. See [8]. Lemma.. Let A be a complex matrix of order. The k (A) k (A) k 1 (A). (4) Equality holds o the left of (4) if ad oly if A is ormal or tra=0. Equality holds o the right of (4) if ad oly if A is ormal.

3 Improvig the localizatio of eigevalues for complex matrices 1859 Proof. See [8]. Disks ad rectagles cotai eigevalues We will use the followig lemma, that is proved i [5]: Lemma.1. If z 1,z,..., z are complex umber, the disk z 1 z i 1 ( z i 1 z i ) give boud for them. Theorem.. Let A be a complex matrix of order with eigevalues λ 1,λ,..., λ. The, all of the eigevalues lie i the disks for j =1,,. λ tra ( 1 )1/ (k j (A) tra )1/, (5) Proof. Let z i = λ i i the lemma.1 ad use (1), () ad (). By lemma., it is trivial that disk (5) for j = is tighter. Later we will give rectagular regios that are cotai eigevalues. Accordig to Schur theorem, there exists a uitary U such that U AU = T = D + N where T is a upper triagular matrix, N is a strictly upper triagular matrix ad D = diag(λ 1,λ,,λ ). Let B = 1 (A + A ) ad C = 1 i (A A ). We compute B, C : B = 1 4 T + T = 1 4 (4 (Rλ i ) + N ), therefore ad similarly B = C = (Rλ i ) + 1 N, (Iλ i ) + 1 N.

4 1860 P. Sargolzaei ad R. Rakhshaipur From the two above relatios, we obtai B C = (Rλ i ) (Iλ i ) = RtrA. O the other had relatio ad (1), (), () ad for j =1,,, we obtai λ i = (Rλ i ) + (Iλ i ). Now, by usig this (Rλ i ) 1 (k j(a)+rtra ) (6) (Iλ i ) 1 (k j(a) RtrA ). (7) Lemma.. If d 1,d,..., d are real umber such that d 1 +d +...+d =0, the d j 1 d k,j =1,,...,. Proof. If d j = 0, we are doe. Suppose x i = d i d j. For i j defie f(x 1,..., x j 1,x j+1,..., x )=1+ x i. Sice x i =1, therefore, f has miimum i j i j i the poit ( 1,..., 1 ), the we have f( 1,..., 1 1 )= 1 1+ i j x i = d i d j. Theorem.4. Let A be a complex matrix of order, with eigevalues λ 1,λ,..., λ. The all of the eigevalues lie i the rectagle where ad [ RtrA α, RtrA α =( 1 )1/ ( β =( 1 )1/ ( + α] [ ItrA ItrA β, + β], (Rλ i ) (RtrA) ) 1/ (8) (Iλ i ) (ItrA) ) 1/. (9)

5 Improvig the localizatio of eigevalues for complex matrices 1861 Proof. The real parts ad imagiary parts of umbers λ 1 tra,..., λ tra, satisfy the coditio of lemma.. Therefore, we apply twice this iequality to obtai (Rλ j RtrA ) 1 (Rλ k RtrA ) = 1 ( (Rλ k ) (RtrA) ) ad (Iλ j ItrA ) 1 1 (Iλ k ItrA ) = 1 ( (Iλ k ) (ItrA) ). The rectagle is icluded i the disk defied (5). Because α + β = ( λ i tra ). Also the vertices of the rectagle are o the boudary disk. Rectagle ca be computed without kowig the eigevalues. Thus, we attai that where ad [ RtrA α j, RtrA α j =( 1 β j =( 1 + α j] [ ItrA for j =1,,. Above rectagle for j = is tighter. β j, ItrA + β j ], (10) )1/ ( 1 (k j(a)+rtra ) (RtrA) ) 1/, (11) )1/ ( 1 (k j(a) RtrA ) (ItrA) ) 1/, (1).1 Mai cotributio: Improved regio I order to cotribute ad provide better regios, we keep goig as follows. Now, we suppose A be a complex matrix with characteristic polyomial havig real coefficiets. Therefore if λ be a eigevalue of A, the λ will be a eigevalue of A too. Thus Iλ i ( 1 (Iλ k ) ) 1/, (1) for i =1,,...,. Take j =, the we obtai Iλ 1 (k (A) tra ). (14) The improved eigevalue localizatio regio will be the itersectio of the rectagular regios with the strip defied above.

6 186 P. Sargolzaei ad R. Rakhshaipur 4 Numerical examples Example 1. Let A = are i 1+i 0 1+i 1 1 4i. Three rectagles of Theorem.4 k 1 =4.0940, Rλ , Iλ 5.66, k =0.774, Rλ , Iλ , k =5.1697, Rλ , Iλ Also three disks Theorem. are λ 1 5 i.8954, λ 1 5 i.6, λ 1 5 i.071. We have show three above rectagles i figure (1) ad last rectagle with last disk i Figure (), (eigevalues was show with *) Figure 1: Rectagles of Example 1

7 Improvig the localizatio of eigevalues for complex matrices Figure : Rectagle ad disk of Example Example. Let B = i i. For this matrix we have 8i 1 0 k (A) =6.57, Rλ.5546, Iλ.951, ad λ I additio, sice characteristic polyomial of B has real coefficiets, by usig strip (14) we obtai Iλ.856. We have show above rectagle ad disk with strip (14) i Figure (): Figure : Rectagle ad disk of Example with strip (14)

8 1864 P. Sargolzaei ad R. Rakhshaipur 5 Cocludig remarks The localizatio of the eigevalues of the complex matrices has studied ad ew improved regios were preseted i this work. For future extesios, we ca suggest that oe ca fid ovel relatios betwee the average ad variace of the eigevalues of the complex matrices to fid the sharper areas cotaiig the eigevalues. Refereces [1] P.J. Eberlei, O measures of o-ormality for matrices, Amer. Math. Moth. 7(1965) [] R. Kress, H.L.De Vries, R. Wegma, O oormal matrices, Liear Algebra Appl. 8(1974) [] O. Rojo, R. Soto, T. Avila, H. Rojo, Localizatio of eigevalues i elliptic regiocs, Comput. Math. Appl. 9(7) (1995) -11. [4] O. Rojo, R. Soto, H. Rojo, New eigevalue estimates for complex matrices, Comput. Math. Appl. 5() (199) -11. [5] O. Rojo, R. Soto, H. Rojo, T.Y. Tam, Eigevalue localizatio for complex matrices, Proyeccioes, 11(1) (199) [6] H. Wolkowicz, G.P.H. Stya, Bouds for eigevalues usig traces, Liear Algebra Appl. 9(1980) [7] H. Wolkowicz, G.P.H. Stya, More bouds for eigevalues usig traces, Liear Algebra Appl. 1(1980) [8] T-Z Huag, L. Wag, Improvig bouds for eigevalues of complex matrices usig traces, Liear Algebra Appl. 46(007) Received: December, 010