Research Article On the Open Problem Related to Rank Equalities for the Sum of Finitely Many Idempotent Matrices and Its Applications

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1 e Scentfc World Journal Volume 204, Artcle ID 70243, 7 pages Research Artcle On the Open roblem Related to Ran Equaltes for the Sum of Fntely Many Idempotent Matrces and Its Applcatons Me-xang Chen,,2 Qng-hua Chen, Qao-xn L, 3 and Zhong-peng Yang 2 School of Mathematcs and Computer Scence, Fujan Normal Unversty, Fuzhou, Fujan , Chna 2 School of Mathematcs, utan Unversty, utan, Fujan 3500, Chna 3 Insttute of Appled hyscs and Computatonal Mathematcs, Bejng 00094, Chna Correspondence should be addressed to Zhong-peng Yang; yangzhongpeng@26com Receved 20 February 204; Revsed 5 June 204; Accepted 8 June 204; ublshed 0 July 204 Academc Edtor: Al Jaballah Copyrght 204 Me-xang Chen et al Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal wor s properly cted Tan and Styan have shown many ran equaltes for the sum of two and three dempotent matrces and ponted out that ran equaltes for the sum + + wth,, be dempotent >3 are stll open In ths paper, by usng bloc Gaussan elmnaton, we obtaned ran equaltes for the sum of fntely many dempotent matrces and then solved the open problem mentoned above Extensons to scalar-potent matrces and some related matrces are also ncluded Introducton Let C m n and GL n C be the sets of m ncomplex matrces and n nnonsngular matrces, respectvely The n ndentty matrx s denoted by I n or smply by I f the sze s mmateral Let Z + be the set of all the postve nteger numbers The symbols ra and A T stand for the ran and transpose of A C m n, respectvely, whle tr A denotes the trace of a square matrx AAmatrxA C n n s sad to be dempotent, f A 2 =A,andscalar-potent determned by λ, f A 2 =λa, for some 0 =λ C see, eg, [ When λ=, t concdes wth the defnton of an dempotent matrx As one of the fundamental buldng blocs n matrx theory, dempotent matrces are very useful n many contexts and have been extensvely studed n the lterature see, eg, [ 6Herewefocusontheresearchontheranofthesum of dempotent matrces Gröss and Trenler have studed ran of the sum of two dempotent matrces see [3, Theorem3Also,Tan and Styan have shown a ran equalty for two dempotent matrces as follows roposton see [,Theorem24and[2,Theorem2 Let, Q C n n be dempotent Then Q r +Q =r rq Q 0 Tan and Styan have extended the ran equalty for the sum of dempotent matrces to the scalar-potent matrces see, eg, [ roposton 2 see [,0 Let, Q C n n be scalar-potent matrces determned by nonzero complexes λ, μthen Q rμ+λq=r Q 0 rq, 2 =λ, Q 2 =μq, λμ=0 Later, Tan and Styan consdered the ran equalty for the sum of three dempotent matrces n [2 as follows roposton 3 see [2, 95 Let, 2, 3 C n n be dempotent Then r =r r r By 3, Tan and Styan have nduced many useful results, for example, f, 2, 3 are dempotent and = 0,then = 2 = 3 = 0 The lteratures [2, 4 6 2

2 2 The Scentfc World Journal show that establshng varous nds of ran equaltes for dempotent matrces s nterestng Tan and Styan ponted out that ran equaltes for the sum + + wth,, be dempotent >3are stll opensee[2,95 In ths paper, by applyng bloc Gaussan elmnaton, ran equaltes for the sum of fntely many dempotent matrces are obtaned These results generalze 3 and solve the open problem proposed by Tan and Styan see, eg, [2 Also, new ran equaltes for fntely many dempotent matrcesaregventheranequalty3 s generalzed to scalar-potent matrces as well 2 Man Results Before showng man results, we need some preparatons Lemma 4 Let A,,A C n m Then A 0 0 A 0 A 2 0 A 2 r d = 0 0 A A A A 2 A 0 = ra +r = A, 4 snce G and S are nonsngular, hence A 0 0 A 0 A 2 0 A 2 r d 0 0 A A A A 2 A 0 =rdag A,,A, = = Ths completes the proof ra +r = A = A The proof method of Lemma 4 s nspred by Marsagla and Styan [5,Theorem 9By4, we get the ranequaltyfor the sum of fntely many dempotent matrces; t s dfferent from the one of three dempotent matrces 3 gven by Tan and Styan Consequently, to fnd the generalzaton of roposton 3 and solve the open problem gven by Tan and Styan see, eg, [2, t s necessary to see a new method dfferent from Lemma 4 Lemma 5 see [7,roblem49 Let C n n be dempotent Then r = tr 7 for any Z + roof Let I n G= d C, 0 0 I n 0 I n I n I n I n I m 0 0 I m 0 I m 0 I m S= d C 0 0 I m I m I m It s evdent that G and S are nonsngular By calculaton, 5 In ths secton, from now on, for A,A 2,,A C n n, one denotes WA,A 2,,A 2 A A 2 A 3 A A = A 2 0 A 2 A 3 A 2 A A 2 A A 3 A 3 A 2 0 A 3 A A 3 A d A A A 2 A A 3 0 A A A A A 2 A A 3 A A 0 C n n Theorem 6 For any Z +,let,, C n n be dempotent Then r = 8 A 0 0 A 0 A 2 0 A 2 G d S 0 0 A A A A 2 A 0 6 =rw,, =rw,, r tr 9 = dag A,,A, = A ; =rw,, tr

3 The Scentfc World Journal 3 roof From Lemma 4,t follows that r d = = r +r = 0 On the other hand, by bloc Gaussan elmnaton, we wll see that r d =rw,, + r In fact, let us wrte the matrx as the quadrparttoned matrx d M M 2 M 2 0 Cn+ n+, 3 Moreover, let then S 2 = 2 E n,0,,0, G 2 =,0,,0, R 2 = 2 T,0,,0 T; I n 0 S 2 I n, I n 0 G 2 I n, I n R 2 are nonsngular By 4and9, we see that X +M 2 G = C n n where M = dag,, C n n, Then by applyng 4and9yelds M 2 = T,,T T C n n, M 2 =,, C n n By and3, t suffces to show 4 Thus, X +M 2 G 2 R 2 +M 2 = 0, T 2,,T T C n n 22 r M M 2 M 2 0 =rw,, + r 5 Drect calculatons to 3showthat I n W 2 M M 2 M 2 0 =M +W 2 M 2 M 2 M 2 0, 6 where W 2 = 0, T 2,, T T C n n, I n W 2 GL n+ C If we defne X =M +W 2 M 2,by4and7, we get X = d S 2 X +M 2 +S 2 M 2 G 2 =S 2 X +M 2 G 2 +M 2 = 2 I,0,, , 2,, =0, 2,, Hence t follows from 4and9that 23 S 2 X +M 2 +S 2 M 2 G 2 R 2 +S 2 M 2 = 2 24

4 4 The Scentfc World Journal Consequently,from 6 24, t follows that I n 0 S 2 I n I n W 2 M M 2 M 2 0 I n 0 G 2 I n I n R 2 = I n 0 S 2 I n X M 2 M 2 0 I n 0 G 2 I n I n R 2 X +M 2 G 2 X +M 2 G 2 R 2 +M 2 = S 2 X +M 2 +S 2 M 2 G 2 S 2 X +M 2 +S 2 M 2 G 2 R 2 +S 2 M d = = d = dag 2,Z, where Z= ,,, 2 Z Z 2 = Z 2 2 C n n 2 26 Snce I n 0 s 2 I n, I n W 2, I n 0 G 2 I n and I n R 2 are nonsngular, we get r M M Z 2 M 2 0 =r Z 2 +r Z Also, t s easy to verfy that 0 I Z n I n 0 Z 2 Z 2 2 I n 0 = 2 Z 2 =W,, Z 2 Z 28 Snce I n 0 and I n 0 are nonsngular, by 3 and 26 28, we obtan r M M 2 M 2 0 =r +rz 29 =r +rw,, Combnng 3 wth29 together wth Lemma 5 yelds the desred results When =3,Theorem6 leads to roposton 3 at once, and when =2,tleadstoroposton; for the dempotent matrces and Q,tfollowsthat 2I n 0 I n 0 W, Q 0 2 I n 30 = 2I n 0 Q I 0 I 2 n Q 0 n 0 0 Q 2 I = n Q 0

5 The Scentfc World Journal 5 For the sum of two dempotent matrces, Tan and Styan have gven out many ran equaltes see [,Theorem 24and [2, Theorems 2, 23, 24, and 27 Let,,, C n n be dempotent; usng Theorem 6 together wth [6, Theorem 6 and [6, 26 yelds the equaltes as follows: r = = r W,, + r r r r d r+ = = r W,,, + r 2 +r d 2 r 2 r d, 3 32 Theorem 6 together wth 3 and32 ndcates that the sum of 3 dempotent matrces has varous nds of ran equaltes, as shown n the dscussons n the lterature [, 2 In vew of, by applyng Lemma 5,weseethefollowng Corollary 7 For any Z +,let,, C n n be dempotent Then r rw,, =r =tr 33 Ths mmedately mples that the dfference of the rans of two bloc matrces n the left sde of 33salwaysequalto r or tr, ndependently on the choce of,when 3 3 The Ran Formulas for the Sum of Scalar-otent Matrces and Applcatons Theorem 6 can easly be extended to scalar-potent matrces; n fact, 2 λ = λ, r λ =r, 34 2 =λ, =,, So /λ s dempotent Theorem 8 For any gven Z +,let C n n be scalarpotent determned by λ =0, =,,Then r [ λ j = j = [ =rw λ, 2,, r =rw λ, 2,, tr λ roof By 8 and34, we get 35 W λ,, λ 2λ λ 2 2 λ 3 3 λ λ λ λ 2 λ λ 2 λ 2 λ 2 λ 2 = λ 3 3 λ 3 λ λ 3 λ 3 λ 3 λ 3 d λ λ λ 2 2 λ λ λ λ λ λ λ 2 2 λ λ 3 3 λ λ 0 36

6 6 The Scentfc World Journal On the other hand, usng 36, we obtan W λ, 2,, =GW λ,, λ G, 37 wth G=dagI n,λ 2 I 2,,λ I n SnceG s nonsngular, by 37, we can wrte rw λ, 2,, =rw λ, λ 2,, 2 λ 38 From 34, /λ s dempotent Usng Theorem 6 together wth 38 yelds the equalty r λ = We note that =rw λ, λ 2 2,, =rw λ, 2,, r = λ =r[ [ j= λ r λ λ j = =r[ λ j = j = [ r λ λ From 39 and40, we get the desred result snce r = r/λ =/λ tr When = 2,ths leads mmedately to roposton 2, sncetcanbewrttenas 2λI n 0 Q I n 0 Q 2λ 0 Q 0 2λ I =, 4 n Q 0 wth 2 =λ, Q 2 =μq,andλμ =0 For any gven dempotent matrx, Farebrother and Trenler [8 denoted the set of generalzed quadratc matrces as Ω n ={A C n n :A 2 =αa+β,a=a=a, α,β C} 42 If =I, t concdes wth the defnton of a quadratc matrx see, eg, [9 In vew of [0,Lemmaand[, Lemma 22, we conclude that 42 can be expressed equvalently as Ω n ={A C n n : A aa b =0, A = A = A, a, b C} If A Ω n,thenfrom42and43, we see that a= 2 α + α 2 +4β, b= 2 α α 2 +4β Lemma 9 For any gven dempotent matrx, fa Ω n satsfes A aa b = 0 wth a =b,thena as a scalar-potent matrx determned by b a roof For the matrx, there exsts a nonsngular matrx S such that =SdagI,0S FromA = A, wecanwrte A = SdagA,A 2 S beng A C FromA = A, we get A 2 = 0;namely,A = SdagA,0S Wehave A ai A bi =0 It s seen from the fact that a matrx s dagonalzable f and only f ts mnmal polynomal has smple roots see [2, Corollary 330 Thus, there exsts a nonsngular matrx S such that A =S dagai r,bi r S Let R=dagS,I;thenRs nonsngular and =Rdag I r,i r,0r, 45 A=Rdag ai r,bi r,0r Hence A a=rdag 0, b a I r,0r = b a R dag 0, I n r,0r Now, t s evdent that A a 2 = b aa a 46 Theorem 0 For any gven dempotent matrces,, C n n and any Z +,fa Ω n satsfes A a A b =0wth a =b, =,,,then r [ b j a j A a = j = [ =rw A b a a,a 2 a 2 2,,A a ra a =rw b a A a,a 2 a 2 2,,A a tr A b a a 47

7 The Scentfc World Journal 7 roof For the dempotent matrces, by applyng Lemma 9, we see that A a s a scalar-potent matrx determned by b a ; then results follow from Theorem 8 [2 R A HornandC R Johnson, Matrx Analyss, Cambrdge Unversty ress, Cambrdge, UK, 985 Conflct of Interests The authors declare that there s no conflct of nterests regardng the publcaton of ths paper Acnowledgments The authors would le to than rofessor Y Tan for hs helpful comments and suggestons on ths paper The authors would also le to than all referees for ther patence n readng ths paper and ther valuable comments and suggestons that are helpful n mprovng and clarfyng the paper Ths wor has been supported by the Natonal Natural Scence Foundaton of Chna grant , the ey tem of Hercynan buldng for the colleges and unverstes servce n Fujan rovnce 2008HX03, and the Specal Scentfc Research rogram n Fujan rovnce Unverstes of Chna Grant JK References [ Y Tan and G H Styan, Ran equaltes for dempotent and nvolutory matrces, Lnear Algebra and Its Applcatons, vol 335, pp 0 7, 200 [2 Y Tan and G H Styan, Ran equaltes for dempotent matrces wth applcatons, Computatonal and Appled Mathematcs,vol9,no,pp77 97,2006 [3 J Gross and G Trenler, Nonsngularty of the dfference of two oblque projectors, SIAM Journal on Matrx Analyss and Applcatons,vol2,no2,pp ,999 [4 W G Cochran, The dstrbuton of quadratc forms n a normal system, wth applcatons to the analyss of covarance, Mathematcal roceedngs of the Cambrdge hlosophcal Socety,vol30,pp78 9,934 [5 GMarsaglaandGHStyan, Equaltesandnequaltesfor rans of matrces, Lnear and Multlnear Algebra, vol2,pp , 974 [6 Y Tan and G H Styan, When does ranabc = ranab + ranbc ranb hold, Internatonal Mathematcal Educaton n Scence and Technology, vol33,pp 27 37, 2002 [7 F Zhang, Matrx Theory: Basc Results and Techeques,Sprnger, New Yor, NY, USA, 999 [8 R W Farebrother and G Trenler, On generalzed quadratc matrces, Lnear Algebra and ts Applcatons,vol40,pp , 2005 [9 M Alesejczy and A Smotunowcz, On propertes of quadratc matrces, Mathematca annonca, vol, no 2, pp , 2000 [0 S Y Lu, Z Yang, and Y Xe, Invarance of ran and nullty for lnear combnatons of generalzed quadratc matrx, Jnln Unversty,vol49,no6,pp ,20 [ Z Yang, X Feng, M Chen, C Deng, and J J Kolha, Fredholm stablty results for lnear combnatons of m-potent operators, Operators and Matrces,vol6,no,pp93 99,202

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