第26卷第1期 2O10年2月 大 学 数 学 CoLLEGE MATHEMATICS Vo126，1 Feb2010 A Series of Upper Bounds for the Perron Root of a Nonnegative Matrix HUGang ，J ING Yah-fel。 (1Schoo1 of Mathematics and Computer Science，Shanxi Normal University，Linfen 04 1 004，China； SchooI of App1Math，University of Electronic Science and Technology of China，Chengdu 610054，China) Abstract：This paPer proposes a series of improved upper bounds for Perron roots of nonegative matrices Better estimates of the Perron root have been given by both diagonal transformation method and the extension of Gerschgorin TheoryNumerica1 experiment has been shown the effectiVeness。f OUr method Key words：nonnegative matrices；Perron roots；diagonal transformation method CLC Number：O15121 Document Code：A Article ID：16721454(2010)01。004303 1 Introduction I et A一(n )nX n be a square nonnegative matrix，it is very well known that A has the maximal nonnegative eigenvalue P(A)，called the Perron root of A，which lies between the smallest and the largest row sum of AMany authors have showed a lot of good bounds for p(A)of the nonnegaUve matrices in the past A sequence of better upper bounds for the Perron root of a nonnegatlve matrlx has been obtained by Tasci and KirklandL through the arithmetic symmetrization of a power of the matrixKoiotilinaE。presented new lower bounds by the geometric symmetrlzatlon oi the matr x and the Rayleigh quotientA new method proposed in Eshas been showed and used for all nonnegatlve irreducible matrices In-this paper，we derive better upper bounds ior the Perron root ot a nonnegattVe matrix through diagonal transformation method and the extension of Gerschgorin Theory Denote A the nonegative matrix，and r1，r2，r its rOW sumsfrom Frobenius theorywe have rp(A)R， where rminr，Rmax- ，iColumn sunis f1，c2，c have i i showed that m n(吉骞 )P(A maX 1 t=l )， which still holds for column sums 1，C2，C For positive matrices，Ledermann ，Ostrowski 。 and Braue r improved(1)successively However，these bounds are special cases of nonnegative matrices 2 Preliminaries For the nonnegative matrix A=A0一(n ) ，set Received date：2007-06-05 、 1 (C n M ) l ( o t t U S e r r a l m S a 44 大 学 数 学 第26卷 “ ) i一1，2，n。 一 一x(耋。 ) (耋。 )卜 ， Lemma 1 。 Let A：(口 )0” ，0Eo，】，then l0(A)maxfQ i+ 一 )， where R一 J，c 一 Lemma 2 Assume A=A。=(以 ) a nonnegative matrix，then |。(A)f。 Note 1 If r 。r，10(A) exists as we11 In this paper，we mainly have an investigation in the case when A iterative proeedure is established by means of diag。na1 transf。rmation。 Ao：A， (3) (4) lS nonsymmetric，to which an Ak+ 一D AD ，D =diag(d，d k， )， 一maxd max(。 )( k)， =1，2，刀， ere ( ao。)( )一 s the parameter, l，2， 上he aboVe transforma ion generates a se of similar matrices A女 and the corresp0nding d e d upp rboundsde“ned aboVeThe c。unterpart of the upper bounds in E3seems as our 3 Main results The。rem 3 Den。te AA。一(以 o) the n。nnegative matrix，and lD(A)its Perr。n r。t The following inequalities hold l0(A) t 。， kE Proof By the transformation A +l=Dr A D ，we have ( a ) (事 ) 一(骞象 ) (耋茅 )卜s， since象and dt are reciprocal and b。th and are positive,。ith 。 篑i bound to be greater than。 。qu j tO unityTherefore，the following tWO cases are considered： case 篑 ， (骞 n：) (砉茅ZA_ k) (骞 ) (砉aj i) s ( ) ( ) 一 ctk)S(d case 2筹 鬟1 If there exists certain parameter (0 1)such that( 2 s( )1-,t kthPn wp h 。 一 、，-I， O 0 ， Il、 ，J-t、 l_ O 第1期 HU Gang，et a1：A Series of Upper Bounds for the Perron Root of a Nonnegative Matrix 45 一(骞l d iin ) (骞务 )h(喾n：) (喾筹n ) ( ) (骞“ l-s(dk)l-, (d )。 (1k) We still have t“ t ，which completes the proof in addition with Lemma 2 Note 2 As a matter of fact，there indeed exists certain parameter ，such that( ) 一(z ) 一 t t when S tends to unity 4 Numerical experiment Problem 1 ASSume A=： l 1 2 2 1 3 2 3 5 It is easy t。get lD(A)一 75311By The。rem 3，let 509999， j10(A)，then we get 1 2 3 4 5 6 7 8 1 2 3 4 5 t (s一0999) 1OOOO 76O5O 75466 75332 7533O t ( 一09999) 10000 76005 75455 75316 753l3 -References Tasci D，Kirkland SA sequence of upper bounds for the Perron root of a nonnegative matrixJLinear Algebra App1，1998，273：2328 Kolotilina L YLower bounds for the Perron root of a nonnegative matrixJLinear Algebra App1，1993，180： 133151 Duan F，Zhang KAn algorithm of diagonal transformation for Perron root of nonnegative irreducible matricesJ App1MathComput，2006，175：762772 Minc HNonnegative MatricesMNew York： ，1998：11 19，2436 Ledermann WBounds for the greatest latent root of a positive matrixEJJLondon MathSoc，1950，25：265 268 OstrowskiBounds for the greatest latent root of a positive matrixJJIondon MathSoc，1952，27：253 256 BrauerThe theorems of I edermann and Ostrowski on positive matricesJDuke MathJ，1957，24：265274 Huang T z，Zhong S M and Li Z I Matrix TheoryMBeijing：Advanced Educational Press，2003：144 非负矩阵Perron根上界序列