Proc. Idia Acad. Sci. Math. Sci.) Vol. 5, No., February 05, pp. 03. c Idia Academy of Scieces Geeralizatio of Samuelso s iequality ad locatio of eigevalues R SHARMA ad R SAINI Departmet of Mathematics, H. P. Uiversity, Shimla 7 005, Idia Correspodig author. E-mail: rajesh_hpu_math@yahoo.co.i MS received 7 September 03; revised 9 October 03 Abstract. We prove a geeralizatio of Samuelso s iequality for higher order cetral momets. Bouds for the eigevalues are obtaied whe a give complex matrix has real eigevalues. Likewise, we discuss bouds for the roots of polyomial equatios. Keywords. Maximum deviatio; cetral momets; Hermitia matrix; eigevalues; coditio umber; polyomial; roots. 00 Mathematics Subject Classificatio. 60E5, 5A, D0.. Itroductio Let x,x,...,x deote real umbers. Their arithmetic mea is the umber x = x i.) ad the r-th cetral momet is m r = x i x) r..) The, s = m is the stadard deviatio of x i i =,,...,). The Samuelso iequality [0] says that or equivaletly s xj x ),.3) x s x j x + s..) The Samuelso iequality also provides a upper boud for the maximum deviatio d from the mea d s,.5) 03
0 R Sharma ad R Saii where d = max x i x..6) i Refiemets, extesios ad applicatios of such iequalities have bee studied extesively i the literature see [,, 8,, ]). Oe refiemet of the Samuelso iequality.5) gives a upper boud for the maximum deviatio i a give rage r = M m, where m x i M,i =,,...,.Thus if s r, the d r + r ) s ad if s r, the d r ) + s r. see []). Such iequalities are useful i various cotexts. Wolkowicz ad Stya [] have observed that if the eigevalues of a complex matrix are real, as i the case of Hermitia matrices, the iequalities.) provide bouds for the extreme eigevalues. Let λ i be eigevalues of A,i =,,...,. Let B = A tra I, where tra deotes the trace of A. The, tra trb λ i tra + trb..7) I some differet cotext ad otatios, iequalities related to.) also appeared i the work of Laguerre [6]. I particular, let x,x,...,x deote the roots, all of which we assume to be real, of the -th degree moic polyomial equatio with, The [6], f x) = x + a x + +a x + a = 0..8) a a + ) a ) a x i ) a ) a.9) for all x i. It is atural to cosider the geralizatio of Samuelso s iequality for higher order momets ad look for related extesios ad applicatios. We obtai a geeralizatio of the Samuelso iequality that gives a lower boud for the cetral momet m ad bouds for all x i i terms of x ad m Theorem. ad Corollary., below). This also provides a upper boud for the maximum deviatio i terms of cetral momet m Corollary.3). As a applicatio we fid bouds for the eigevalues of a matrix whe all its eigevalues are real Theorem 3.). A upper boud for the coditio umber is give i Corollary 3.. We give examples ad compare our bouds with those give by Wolkowicz ad Stya []. Likewise, we give bouds for the largest ad smallest roots of a polyomial equatio whe all its roots are real Theorem.).
. Mai results Geeralizatio of Samuelso s iequality 05 Theorem.. If m is the cetral momet of real umbers x,x,...,x, the + ) m xj ) x ) for all j =,,..., ad r =,,....) Proof. From eq..), we write xj x ) m = +,i =j x i x)..) For m positive real umbers y i, i =,,...,m, ) k m yi k m y i,k=,,..3) m m Applyig.3) to positive real umbers x i x),i =,,...,ad i = j, we get,i =j x i x) ) r,i =j r x i x)..) Further, the Cauchy Schwarz iequality implies that the iequality.3) holds good for all real umbers y i whe k =. Therefore,i =j x i x),i =j x i x)..5) O the other had, the sum of all the deviatios from the mea is zero, therefore x i x) = 0, ad we get that,i =j Combiig.).6), we fid that x i x) = x x j..6),i =j ) x x i x) xj..7) Isertig.7) i.), ad by computig leads to.).
06 R Sharma ad R Saii It is clear from the derivatio that iequality.) is strict if x j is differet from m or M. Equality holds i.) whe x j = m ad x i = M or x j = M ad x i = m,i =,,..., ad i = j. We ow fid bouds for x j i terms of mea x) ad cetral momet m ) i the followig corollary. COROLLARY. With otatios as above, x for all j =,,...,. ) + ) m Proof. It follows from iequality.) that xj x ) ) + ) m ) x j x + ) + ) m ).8) ) r..9) Sice, y a if ad oly if a y a, the iequalities.8) follow from.9). Equality holds o the leftright) of.8) if ad oly if largestsmallest) x i are all equal. The upper boud for the maximum deviatio d from the mea is give i the followig corollary. COROLLARY.3 If d is a maximum deviatio from the mea x) of real umbers x,x,...,x, the d ) + ) m )..0) Proof. The iequality.9) is evidetly equivalet to.0). Note that Samuelso s iequalities.3),.) ad.5) are respectively the special cases of iequalities.),.8) ad.0), r =. Oe of the iterests i the special case r = is that the sample kurtosis is defied i terms of the secod ad fourth cetral momet as β = m m. For r =, the iequality.8) is tighter tha.). This follows from the fact that [3] β 3 + 3. For r = ad = 3, the iequalities.) ad.8) give equal estimates. It also follows that if the value of the fourth cetral momet m ) is prescribed, the.0) gives a
Geeralizatio of Samuelso s iequality 07 better boud for maximum deviatio tha the correspodig boud give by Samuelso s iequality.5). 3. Bouds for eigevalues usig traces Let A be a complex matrix with real eigevalues as i case of a Hermitia matrix. Several iequalities for the eigevalues i terms of traces of A,A ad A 3 are give i the literature see [, ]). It is costly to calculate the traces of higher powers of A. However, it is always of iterest to kow if better estimates ca be obtaied at the cost of more calculatios. We obtai here a geeralizatio of iequality.7) that ivolves a trace of B, r =,,... ad we metio a umber of examples to illustrate the utility of these bouds. The coditio umber of a positive defiite matrix is the ratio of the largest to the smallest eigevalue. The bouds for this ratio i terms of the traces are give i []. We show by meas of examples that our bouds are also useful i estimatig the upper limit of the coditio umber. Theorem 3.. Let A be a complex matrix with real eigevalues λ i, i =,,..., ad let B = A tra I. The, ) tra ) ) trb λ i tra + ) + ) trb + ) 3.) for all i =,,...,,ad r =,, 3,... Proof. The arithmetic mea of eigevalues λ,λ,...,λ is x = λ i = tra. 3.) The eigevalues of the matrix B are λ i tr A. Therefore m = λ i tra ) = trb. 3.3) The iequality 3.) follows from.8), ad substitute the values of x ad m from 3.) ad 3.3), respectively. Let λ i be the eigevalues of a Hermitia matrix A, ad suppose m λ i M,i =,,...,. Wolkowicz ad Stya [] have show that if tr A>0 ad p = tra) tra ) > 0, 3.) the A is positive defiite ad M m + p ). 3.5) p
08 R Sharma ad R Saii The iequality 3.5) gives a upper boud for the coditio umber ) M m.but,for example, for the matrix 5 A = 6, 3.6) 8 the value of p< 0 ad therefore 3.5) is ot applicable. The followig corollary is stroger tha 3.5). COROLLARY 3. Let A be a Hermitia matrix. If tra>0 ad ) tra ) trb 3.7) + ) for some r =,, 3,..., the A is positive defiite ad M m β α, 3.8) where α ad β are respectively the lower ad upper bouds i 3.). Proof. The assertios of the corollary follow immediately from Theorem 3.. If 3.7) holds, the first iequality 3.) shows that all the eigevalues of A are positive. The matrix A i 3.6) satisfies the coditio of Corollary 3. for r = 3, therefore M m 55.8. This also shows that 3.7) is more useful tha 3.). We ow metio several examples to illustrate the effectiveess of the bouds i Theorem 3.. Example. For the 3 3 matrix, 5 i A = + i, + i i 3 the estimates of Wolkowicz ad Stya [] ad 3.) with r = both give.7 λ i 7.7. For r = 3, we have from 3.),.35 λ i 7.35. The eigevalues of A are 3, 3 ± 5. We ow borrow examples of Wolkowicz ad Stya [] to compare the bouds 3.5) ad 3.8) for the coditio umber of a positive defiite matrix. Example. Let A = 0 3 0 5 0 0 6 0 3 0 7.
Geeralizatio of Samuelso s iequality 09 The estimate 3.5) gives M m 3.98 see []), while from 3.8), M m.600, r =. Example 3. Let 5 A = 6 7. 8 From 3.5), M m 36.973 see []), while from 3.8), M m 8.98,r =.. Bouds o roots of polyomials I the theory of polyomial equatios, the study of polyomials with real roots is of special iterest see [7, 9]). It is also of iterest to fid bouds o the roots i terms of the coefficiets see [6, ]). Oe such boud is the Laguerre iequality.9) which gives a estimate for the roots of the polyomial eq..8) i terms of its first two coefficiets a ad a. It is sufficiet to cosider the polyomial equatio i which the coefficiet of x is zero, f x) = x + b x + b 3 x 3 + +b x + b = 0..) The eq..) is obtaied o dimiishig the roots of.8) by a. The the Laguerre iequality says that all the roots of.) lie i the iterval [ D,D ], where D = )b. We show that better bouds ca be give if we ivolve other coefficiets of the eq..). Let x,x,...,x be the roots of.). O usig the well-kow Newto s idetity α k + b α k + b α k + +b k α + kb k = 0, where α k = xi k ad k =,,...,.Wehave m = 0, m = xi = b,m = ad m 6 = xi = b b ).) xi 6 = b3 + 3b 3 + 6b b 6b 6..3) Usig such relatios i iequality.8) we ca obtai better estimates for the roots of eq..). As a example, we metio here a improvemet of Laguerre s iequality i the followig theorem. Theorem.. All the roots of the polyomial equatio.) with 5 lie i the iterval [ D,D ], where ) ) 3 D = + ) 3 b b ).
0 R Sharma ad R Saii Proof. From.8), for m = x = 0 ad r =, we have ) ) 3 ) + ) 3 m ) 3 x j + ) 3 m..) Substitutig the value of m from.) i.), we fid that xj D. Example. Let fx) = x 5 + 5x + x 3 + 96x + x + = 0..5) If all the coefficiets a i s i the polyomial eq..8) are positive ad a i a i+a i > 0, i =,,...,, the all its roots are real see [5]). So, all the roots x i of.5) are real, i =,,...,5.Let y i = x i 5 be the roots of the dimiished equatio fy) = y 5 38y 3 + 96y 9y + 33 = 0. The Laguerre iequality.9) gives y i.859 while from Theorem., y i.57. Example 5. Let fx) = x 5 + 80x + 500x 3 + 5000x + 3750x + = 0..6) 5 The roots of f x) are real. Let y = x 6, the.6) gives fy) = y 5 060y 3 + 90y + 70y 36879 5 From.9), y i.83 while from Theorem., y i 38.38. = 0. Ackowledgemets The authors are grateful to Prof. Rajedra Bhatia for useful discussios ad suggestios, ad to ISI, Delhi for a visit i Jauary 03 whe this work had begu. The support of the UGC-SAP is ackowledged. Refereces [] Bhatia R, Matrix aalysis 000) New York: Spriger Verlag) [] Bhatia R ad Davis C, A better boud o the variace, Amer. Math. Mothly 07 000) 353 357 [3] Dale J, Algebraic bouds o stadardized sample momets, Statist. Prob. Lett. 5 987) 39 33 [] Jese S T ad Stya G P H, Some commets ad a bibliography o the Laguerre Samuelso iequality with extesios ad applicatios i statistics ad matrix theory, i: Aalytic ad Geometric Iequalities ad applicatios eds) H M Srivastava ad T M Rassias 999) Kluwer) pp. 5 8
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