An S-type upper bound for the largest singular value of nonnegative rectangular tensors

Transcription

1 Ope Mat Ope Matematics Ope Access Researc Article Jiaxig Za* ad Caili Sag A S-type upper bud r te largest sigular value egative rectagular tesrs DOI 0.55/mat Received August 3, 06 accepted Octber, 06. Abstract A S-type upper bud r te largest sigular value a egative rectagular tesr is give by breakig N D g it disit subsets S ad its cmplemet. It is sw tat te ew upper bud is smaller ta tat prvided by Yag ad Yag (0). Numerical examples are give t veriy te teretical results. Keywrds Negative tesr, Rectagular tesr, Sigular value MSC 5A8, 5A4, 5A69 Itrducti Let R.C/ be te real (cmplex) ield, p q m be psitive itegers, m, N D g, ad R C be te ce x D.x x x / T R W x i 0 i N g. A real.p q/-t rder m dimesial rectagular tesr, r simply a real rectagular tesr A is deied as llws A D.a i i p q / were a i i p q R r i k D m k D p ad k D k D q We p D q D A is simply a real m rectagular matrix. Tis ustiies te wrd rectagular". Fr ay vectr x ad ay real umber, dete x Œ D.x x x /T Let Ax p y q be a vectr i R m suc tat m.ax p y q / i D a ii i p q x i x ip y y q were i D m. Similarly, let Ax p y q.ax p y q / D i i p D q D m be a vectr i R suc tat i i p D q D a i i p q x i x ip y y q were D Let l D p C q. I tere are a umber C vectrs x C m 0g, ad y C 0g suc tat ( Ax p y q D x Œl Ax p y q D y Œl te is called te sigular value A, ad.x y/ is te let ad rigt eigevectrs pair A, assciated wit, respectively. I R x R m ad y R, te we say tat is a H-sigular value A, ad.x y/ is te let *Crrespdig Autr Jiaxig Za Cllege Sciece, Guizu Mizu Uiversity, Guiyag 55005, Cia, zx8004@63.cm Caili Sag Cllege Sciece, Guizu Mizu Uiversity, Guiyag 55005, Cia, sagcl@6.cm 06 Za ad Sag, publised by De Gruyter Ope. Tis wrk is licesed uder te Creative Cmms Attributi-NCmmercial-NDerivs 3.0 Licese. Uauteticated Dwlad Date 8/5/8 65 AM

2 96 J. Za, C. Sag ad rigt H-eigevectrs pair assciated wit, respectively. I a sigular value is t a H-sigular value, we call it a N-sigular value A. I p D q D, te tis is ust te usual deiiti sigular values r a rectagular matrix []. We call 0 D W is a sigular value Ag is te largest sigular value []. Nte ere tat te ti sigular values r tesrs was irst prpsed by Lim i [3]. We l is eve, te deiiti i [] is te same as i [3]. We l is dd, te deiiti i [] is sligtly dieret rm tat i [3], but parallel t te deiiti eigevalues square matrices [4] see [] r details. Fr te sake simplicity, te deiiti sigular values i tis paper is te deiiti i []. We recall te weak Perr-Frbeius terem r egative rectagular tesrs, wic was give i []. Terem. ([, Terem ]). Let A be a.p q/-t rder m dimesial egative tesr. Te 0 is te largest sigular value wit egative let ad rigt eigevectrs pair.x y/ R m C 0g R C0g crrespdig t it. Te largest sigular value a egative rectagular tesr as a wide rage practical applicatis i te strg ellipticity cditi prblem i slid mecaics [5, 6] ad te etaglemet prblem i quatum pysics [7, 8]. Recetly, tere are may results abut te prperties square tesrs, especially te upper buds r te Z-spectral radius ad H-spectral radius a egative square tesr [9 3]. Hwever, tere are results abut te upper buds r te largest sigular value a egative rectagular tesr except te llwig e []. Terem. (see [, Terem 4]). Let A be a.p q/-t rder m dimesial egative rectagular tesr. Te mi R i C g 0 R i C g im im were R i D m i i p D q D a ii i p q C D m i i p D q D a i i p q Trugut tis paper, we assume m D. Our gal i tis paper is t give a ew upper bud r te largest sigular value a egative rectagular tesr, ad prve tat te ew upper bud is smaller ta tat i Terem.. Mai results We begi wit sme tati. Give a empty prper subset S N, we dete ad te i ip q N ı ii ip q D0 N WD.i i p q / W i i p q N g S WD.i i p q / W i i p q Sg N WD.i i p q / W i i p q N g S WD.i i p q / W i i p q Sg S D N S S D N S Tis implies tat r a egative rectagular tesr A D.a i i p q / we ave tat r i S, r i D a D r S iiipq i C r S i r S D r i i C r S i a i c D i ip q N ı i ip q D0 a iipq D c S C c S c i D cs C c S a ii ii Uauteticated Dwlad Date 8/5/8 65 AM

3 A S-type upper bud r te largest sigular value egative rectagular tesrs 97 were ad r S i D c S D ı i i p q D.i ip q / S ı ii ip q D0.i ip q / S ı i ip q D0 ( i i D D i p D D D q 0 terwise a iiipq r a iipq c S i S D D.i i p q / S a ii i p q.i i p q / S a i i p q By Terem., te llwig lemma is easily btaied. Lemma.. Let A be a.p q/-t rder dimesial egative rectagular tesr. Te 0 a iiii i N Terem.. Let A be a.p q/-t rder dimesial egative rectagular tesr, S be a empty prper subset N, S be te cmplemet S i N. Te 0 U S D U S U S U S U S g were U S D U S D U S D U S D is S a iiii C a C r S C.a iiii a r S is S a iiii C a C r S C.a iiii a r S is S a iiii C a C c S C.a iiii a c S is S a iiii C a C c S / C 4 r i c i gr S / C 4 r i c i gr S / C 4 r i c i gc S C.a iiii a c S / C 4 r i c i gc S Pr. Let 0 be te largest sigular value A. Accrdig t Terem., tere exist tw zer egative vectrs x D.x x x / T ad y D.y y y / T suc tat ( Ax p y q D 0 x Œl () Let Ax p y q D 0 y Œl () x t D x i W i Sg x D x i W i SgI y D y i W i Sg y g D y i W i SgI w i D x i y i g i N w S D w i W i Sg w S D w i W i Sg Te, at least e x t ad x is zer, ad at least y ad y g is zer. Next, we preset ur cases t prve tat. Uauteticated Dwlad Date 8/5/8 65 AM

4 98 J. Za, C. Sag Case I Suppse tat w S D x t w S D x, te x t y t x y. (i) I x x t, te x D w i W i N g. By te -t equality i (), we ave. 0 a /x l 0 x l a x p y q D a i i p q x i x ip y y q.i i p q / S C a i i p q x i x ip y y q Hece,.i ip q / S ı i ip q D0 a i i p q x l t.i i p q / S D r S xl t C r S xl C.i ip q / S ı i ip q D0 a i i p q x l. 0 a r S /xl r S xl t (3) I x t D 0, te 0 a r S 0 as x > 0 ad it is bvius tat 0 U S. Oterwise, x t > 0. Frm te t-t equality i (), we ave i.e.,. 0 a tttt /xt l 0 xt l a tttt x p t y q t D a x x y y tiipq i ip q i ip q N ı ti ip q D0 i ip q N ı ti ip q D0 D r t x l a x l tiipq. 0 a tttt /x l t r t x l (4) Frm Lemma., we ave 0 a tttt 0. Multiplyig (3) wit (4), we ave. 0 a tttt /. 0 a r S /xl t x l r t r S xl t x l Nte tat xt l x l > 0. Te is S. 0 a tttt /. 0 a r S Slvig (5) gives 0 a tttt C a C r S C.a tttt a r S a iiii C a C r S U S C.a iiii a r S / r tr S (5) / C 4r t r S (ii) I x t x, te x t D w i W i N g. Similarly t te pr (i), we ca btai tat. 0 a /. 0 a tttt r S t / r rt S / C 4r i r S Tis gives 0 a C a tttt C rt S C.a a tttt rt S / C 4r rt S Uauteticated Dwlad Date 8/5/8 65 AM

5 is S U S A S-type upper bud r te largest sigular value egative rectagular tesrs 99 a iiii C a C r S C.a iiii a r S / C 4r i r S Case II Suppse tat w S D y w S D y g, te y x y g x g. I y g y, te y g D w i W i N g. Similarly t te pr (i) i Case I, we ave is S. 0 a /. 0 a gggg c S g / c cg S Tis gives 0 a C a gggg C cg S C.a a gggg cg S a iiii C a C c S U S C.a iiii a c S I y y g, te y D w i W i N g. Similarly t te pr (ii) i Case I, we ave is S. 0 a gggg /. 0 a c S / c gc S Tis gives 0 a gggg C a C c S C.a ggg a c S a iiii C a C c S U S C.a iiii a c S / C 4c cg S / C 4c i c S / C 4c g c S / C 4c i c S Case III Suppse tat w S D x t w S D y g, te x t y t y g x g. I y g x t, te y g D w i W i N g. Similarly t te pr (i) i Case I, we ave is S. 0 a tttt /. 0 a gggg c S g / r tcg S Tis gives 0 a tttt C a gggg C cg S C.a tttt a gggg cg S a iiii C a C c S U S C.a iiii a c S / C 4r t cg S I x t y g, te x t D w i W i N g. Similarly t te pr (ii) i Case I, we ca btai tat. 0 a gggg /. 0 a tttt r S t / c g rt S Tis gives 0 a gggg C a tttt C rt S C.a gggg a tttt rt S a iiii C a C r S is S U S C.a iiii a r S / C 4r i c S / C 4c g rt S / C 4c i r S Case IV Suppse tat w S D y w S D x, te y x x y. I x y, te x D w i W i N g. Similarly t te pr (i) i Case I, we ave. 0 a /. 0 a r S / c r S Uauteticated Dwlad Date 8/5/8 65 AM

6 930 J. Za, C. Sag Tis gives 0 a C a C r S is S U S a iiii C a C r S C.a a r S C.a iiii a r S / C 4c r S / C 4c i r S I y x, te y D w i W i N g. Similarly t te pr (ii) i Case I, we ca btai tat Tis gives 0. 0 a /. 0 a c S a C a C c S is S U S a iiii C a C c S Te cclusi llws rm Case I, II, III ad IV. / r c S C.a a c S C.a iiii a c S Next, we cmpare te upper bud i Terem. wit tat i Terem.. / C 4r c S / C 4r i c S Terem.3. Let A be a.p q/-t rder dimesial egative rectagular tesr, S be a empty prper subset N, S be te cmplemet S i N. Te Pr. Here, we ly prve U S i N U S R i C g i N R i C g Similarly, we ca prve U S U S U S i N R i C g respectively. We ext divide it tw cases t prve. Case I Suppse tat U S D is S a iiii Ca Cr S (i) Fr ay i S S i R i R te Hece, Furtermre,.a iiii a r S C.a iiii a r S 0 r i a a iiii C r S / C 4r i r S C r S.a iiii a r S / C 4.a a iiii C r S C r S / C4r i r S /r S D.a iiii a r S / C 4.a a iiii C r S /r S C 4.r S / D.a a iiii C r S C r S / a iiii C a C r S C Œ.a iiii a r S / C 4r i r S a iiii C a C r S C Œa a iiii C r S C r S D a C r S C r S Uauteticated Dwlad Date 8/5/8 65 AM

7 A S-type upper bud r te largest sigular value egative rectagular tesrs 93 wic implies U S D D R is S a iiii C a C r S R g R i C g S i N (ii) Fr ay i S S i R R i te Similarly t te pr (i), we ca btai Hece, C.a iiii a r S 0 r S a iiii a r S C r i / C 4r i r S a iiii C a C r S C Œ.a iiii a r S / C 4r i r S Ri U S D is S a iiii C a C r S R i g R i C g is i N Case II Suppse tat U S D is S a iiii Ca Cr S (iii) Fr ay i S S i C i R te Similarly t te pr (i), we ca btai C.a iiii a r S C.a iiii a r S 0 c i a a iiii C r S C r S / C 4r i r S / C4c i r S a iiii C a C r S C Œ.a iiii a r S / C 4c i r S R Hece, U S D is S a iiii C a C r S R g R i C g S i N (iv) Fr ay i S S i R C i te Similarly t te pr (ii), we ca btai Hece, C.a iiii a r S 0 r S a iiii a r S C c i / C 4c i r S a iiii C a C r S C Œ.a iiii a r S / C 4c i r S Ci U S D is S a iiii C a C r S C i g R i C g is i N Te cclusi llws rm Cases I ad II. C.a iiii a r S / C 4c i r S Uauteticated Dwlad Date 8/5/8 65 AM

8 93 J. Za, C. Sag 3 Numerical examples I tis secti, tw umerical examples are give t veriy te teretical results. Example 3.. Let A D.a ikl / be a. /-t rder 3 3 dimesial egative rectagular tesr wit etries deied as llws A.W W / D A.W W / D A.W W 3 / D A.W W / D A.W W / D A.W W 3 / D A.W W 3/ D A.W W 3/ D A.W W 3 3/ D Let S D g Obviusly S D 3g By Terem., we ave 0 45 By Terem., we ave I act, 0 D 3378 Tis example sws tat te upper bud i Terem. is smaller ta tat i Terem.. Example 3.. Let A D.a ikl / be a. /-t rder dimesial egative rectagular tesr wit etries deied as llws a D a D a D a D a D a D ter a ikl D 0 By Terem., we ave 0 3 I act, 0 D 3 Tis example sws tat te upper bud i Terem. is sarp. 4 Cclusis I tis paper, by breakig N it disit subsets S ad its cmplemet, a S-type upper bud U S r te largest sigular value a egative rectagular tesr A wit m D is btaied, wic imprves te upper bud i []. Te a iterestig prblem is w t pick S t make U S as small as pssible. But tis is diicult we is large. I te uture, we will researc tis prblem. Ackwledgemet Te autrs are very idebted t te reviewers r teir valuable cmmets ad crrectis, wic imprved te rigial mauscript tis paper. Tis wrk is supprted by te Natial Natural Sciece Fudati Cia (Ns.36074,504), Fudati Guizu Sciece ad Teclgy Departmet (Grat N.[05]073) ad Natural Sciece Prgrams Educati Departmet Guizu Prvice (Grat N. [06]066). Uauteticated Dwlad Date 8/5/8 65 AM

9 A S-type upper bud r te largest sigular value egative rectagular tesrs 933 Reereces [] Cag K.C., Qi L.Q., Zu G.L., Sigular values a real rectagular tesr, J. Mat. Aal. Appl., 00, 370, [] Yag Y.N., Yag Q.Z., Sigular values egative rectagular tesrs, Frt. Mat. Cia, 0, 6(), [3] Lim L.H., Sigular values ad eigevalues tesrs A variatial apprac, Prceedigs te IEEE Iteratial Wrksp Cmputatial Advaces i Multi-Sesr Adaptive Prcessig (CAMSAP 05), 05,, 9-3 [4] Cag K.C., Pears K., Zag T., O eigevalue prblems real symmetric tesrs, J. Mat. Aal. Appl., 009, 350, 46-4 [5] Kwles J.K., Sterberg E., O te ellipticity te equatis -liear elaststatics r a special material, J. Elasticity, 975, 5, [6] Wag Y., Ar M., A rermulati te strg ellipticity cditis r ucstraied yperelastic media, J. Elasticity, 996, 44, [7] Dal D., Leiass J.M., Myreim J., Ovrum E., A tesr prduct matrix apprximati prblem i quatum pysics, Liear Algebra Appl., 007, 40, 7-75 [8] Eistei A., Pdlsky B., Rse N., Ca quatum-mecaical descripti pysical reality be csidered cmplete?, Pys. Rev., 935, 47, [9] Li C.Q., Li Y.T., Kg., New eigevalue iclusi sets r tesrs, Numer. Liear Algebra Appl., 04,, [0] Li C.Q., Ce Z., Li Y.T., A ew eigevalue iclusi set r tesrs ad its applicatis, Liear Algebra Appl., 05, 48, [] Li C.Q., Jia A.Q., Li Y.T., A S-type eigevalue lcalizati set r tesrs, Liear Algebra Appl., 06, 493, [] He J., Huag T.Z., Upper bud r te largest Z-eigevalue psitive tesrs, Appl. Mat. Lett., 04, 38, 0-4 [3] Li W., Liu D.D., Vg S.W., Z-eigepair buds r a irreducible egative tesr, Liear Algebra Appl., 05, 483, 8-99 Uauteticated Dwlad Date 8/5/8 65 AM