An S-type singular value inclusion set for rectangular tensors

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1 Sang Jounal of Inequaliies and Applicaions 2017) 2017:141 DOI /s R E S E A R C H Open Access An S-ype singula value inclusion se fo ecangula ensos Caili Sang * * Coespondence: sangcl@126.com College of Daa Science and Infomaion Engineeing, Guizou Minzu Univesiy, Guiyang, Guizou , P.R. Cina Absac An S-ype singula value inclusion se fo ecangula ensos is given. Based on e se, new uppe and lowe bounds fo e lages singula value of nonnegaive ecangula ensos ae obained and poved o be sape an some exising esuls. Numeical examples ae given o veify e eoeical esuls. MSC: 15A18; 15A69 Keywods: ecangula ensos; nonnegaive ensos; singula value; inclusion se 1 Inoducion Le RC) be e eal complex) field, p, q, m, n be posiive ineges, l = p + q, m, n 2and N = 1,2,...,n}.WecallA =a i1 i p 1 q )aealp, q) ode m n dimensional ecangula enso, o simply a eal ecangula enso, denoed by A R [p,q;m,n],if a i1 i p 1 q R, 1 i 1,...,i p m,1 1,..., q n. Wen p = q =1,A is simply a eal m n ecangula maix. Tis usifies e wod ecangula. We call A nonnegaive, denoed by A R [p,q;m,n] +,ifeacofiseniesa i1 i p 1 q 0. Fo any vecos x =x 1, x 2,...,x m ) T, y =y 1, y 2,...,y n ) T and any eal numbe α, denoe x [α] =x α 1, xα 2,...,xα m )T and y [α] =y α 1, yα 2,...,yα n )T.LeAx p 1 y q be a veco in R m suc a Ax p 1 y q) i = m n i 2,...,i p =1 1,..., q =1 a ii2 i p 1 q x i2 x ip y 1 y q, wee i =1,...,m. Similaly, le Ax p y q 1 be a veco in R n suc a Ax p y q 1) = m n i 1,...,i p =1 2,..., q =1 a i1 i p 2 q x i1 x ip y 2 y q, wee =1,...,n. Ifeeaeanumbeλ C, vecosx C m \0}, andy C n \0} suc a Ax p 1 y q = λx [l 1], Ax p y q 1 = λy [l 1], Te Auos) Tis aicle is disibued unde e ems of e Ceaive Commons Aibuion 4.0 Inenaional License p://ceaivecommons.og/licenses/by/4.0/), wic pemis unesiced use, disibuion, and epoducion in any medium, povided you give appopiae cedi o e oiginal auos) and e souce, povide a link o e Ceaive Commons license, and indicae if canges wee made.

2 Sang Jounal of Inequaliies and Applicaions 2017) 2017:141 Page 2 of 14 en λ is called e singula value of A,andx, y) is a pai of lef and ig eigenvecos of A, associaedwiλ, especively.ifλ R, x R m,andy R n,enwesayaλ is an H-singula value of A,andx, y) is a pai of lef and ig H-eigenvecos associaed wi λ, especively. If a singula value is no an H-singula value, we call i an N-singula value of A [1]. We call λ 0 = max λ : λ is a singula value of A } e lages singula value [2]. Noe ee a e definiion of singula values fo ensos was fis poposed by Lim in [3]. Wen l is even, e definiion in [1]isesameasin[3]. Wen l is odd, e definiion in [1] is sligly diffeen fom a in [3], bu paallel o e definiion of eigenvalues of squae maices [4];see [1] fo deails. Wen m = n, suc eal ecangula ensos ave a sound applicaion backgound. Fo example, e elasiciy enso is a enso wi p = q =2andm = n = 2 o 3; fo deails, see [1]. Due o e fac a singula values of ecangula ensos ave a wide ange of pacical applicaions in e song ellipiciy condiion poblem in solid mecanics [5, 6] and e enanglemen poblem in quanum pysics [7, 8], vey ecenly, i as aaced aenion of eseaces [9 17]. Cang e al. [1] sudied some popeies of singula values of ecangula ensos, wic include e Peon-Fobenius eoem of nonnegaive ieducible ensos. Yang e al.[2] exended e Peon-Fobenius eoem of nonnegaive ieducible ensos o nonnegaive ensos, and gave e uppe and lowe bounds of e lages singula value of nonnegaive ecangula ensos. Ou goal in is pape is o popose a singula value inclusion se fo ecangula ensos and use e se o obain new uppe and lowe bounds fo e lages singula value of nonnegaive ecangula ensos. 2 Main esuls In is secion, we begin wi some noaion. Le A R [p,q;n,n].fo i, N, i,denoe R i A)= a ii2 i p 1 q, i 2,...,i p, 1,..., q N i A)= a ii2 i p 1 q = R i A) a i, δ i2 ip 1 q =0 C A)= a i1 i p 2 q, c i A)= i 1,...,i p, 2,..., q N δ i1 ipi 2 q =0 a i1 i p 2 q = C A) a i ii i, wee 1 ifi 1 = = i p = 1 = = q, δ i1 i p 1 q = 0 oewise.

3 Sang Jounal of Inequaliies and Applicaions 2017) 2017:141 Page 3 of 14 Teoem 1 Le A R [p,q;n,n], S be a nonempy pope subse of N, S beecomplemenof SinN. Ten wee σ A) ϒ S A)= i S, S ˆϒ i, A) ϒ i, A) )) i S, S ˆϒ i, A) ϒ i, A) )), ˆϒ i, A)= z C : z i A)) z a i max R A), C A) }}, ϒ i, A)= z C : z c i A)) z a i max R A), C A) }}. Poof Fo any λ σ A), le x =x 1, x 2,...,x n ) T C n \0} and y =y 1, y 2,...,y n ) T C n \0} be e associaed lef and ig eigenvecos, a is, Ax p 1 y q = λx [l 1], 1) Ax p y q 1 = λy [l 1]. 2) Le x s = max i S xi }, x = max i S w i = max xi, y i }, w S = max w i}, i N i S xi }, y g = max yi }, y = max yi }, i S i S w S = maxw i }. i S Ten, a leas one of x s and x is nonzeo, and a leas one of y g and y is nonzeo. We divide e poof ino fou pas. Case I: Suppose a w S = x s, w S = x,en x s y s, x y. i) If x s x,en x s = max i N w i }.Tes equaliy in 1)is λx l 1 s = a si2 i p 1 q x i2 x ip y 1 y q + a s x p 1 y q. δ i2 ip 1 q =0 Taking modulus in e above equaion and using e iangle inequaliy give λ x s l 1 a si2 i p 1 q x i2 x ip y 1 y q δ i2 ip 1 q =0 + a s x p 1 y q a si2 i p 1 q x s l 1 + a s x l 1 δ i2 ip 1 q =0 = s A) x s l 1 + a s x l 1, i.e., λ s A) ) x s l 1 a s x l 1. 3)

4 Sang Jounal of Inequaliies and Applicaions 2017) 2017:141 Page 4 of 14 If x =0,en λ s A) 0as x s > 0, and i is obvious a λ s A) ) λ 0 a s R A), wic implies a λ ˆϒ s, A). Oewise, x > 0. Moeove, fom e equaliy in 1), we can ge λ x l 1 a i2 i p 1 q x i2 x ip y 1 y q i 2,...i p, 1,..., q N R A) x s l 1. 4) Muliplying 3)by4) and noing a x s l 1 x l 1 >0,weave λ s A) ) λ a s R A), wic also implies a λ ˆϒ s, A) ˆϒ i S, S i, A). ii) If x x s,en x = max i N w i }. Similaly, we can ge λ s A) ) λ a s ss s R s A), and λ ˆϒ,s A) ˆϒ i S, S i, A). Case II: Suppose a w S = y g, w S = y,en y g x g, y x. i) If y g y,en y g = max i N w i }.Teg equaliy in 2)is λy l 1 g = δ i1 ip 2 q =0 a i1 i p g 2 q x i1 x ip y 2 y q + a g x p yq 1. Taking modulus in e above equaion and using e iangle inequaliy give λ y g l 1 a i1 i p g 2 q x i1 x ip y 2 y q δ i1 ip 2 q =0 + a g x p y q 1 a i1 i p g 2 q y g l 1 + a g y l 1 δ i1 ip 2 q =0 = c g A) y g l 1 + a g y l 1, i.e., λ c g A) ) y g l 1 a g y l 1. 5) If y =0,en λ c g A) 0as y g > 0, and fuemoe λ c g A) ) λ 0 a g C A),

5 Sang Jounal of Inequaliies and Applicaions 2017) 2017:141 Page 5 of 14 wic implies a λ ϒ g, A). Oewise, y > 0. Moeove, fom e equaliy in 2), we can ge λ y l 1 a i1 i p 2 q x i1 x ip y 2 y q i 1,...,i p, 2,..., q N C A) y g l 1. 6) Muliplying 5)by6) and noing a y g l 1 y l 1 >0,weave λ c g A) ) λ a g C A), wic also implies a λ ϒ g, A) ϒ i S, S i, A). ii) If y y g,en y = max i N w i }. Similaly, we can ge λ c g A)) λ a g gg g C g A), and λ ϒ,g A) ϒ i S, S i, A). Case III: Suppose a w S = x s, w S = y,en x s y s, y x.if x s y,en x s = max i N w i }. Simila o e poof of 3)and6), we ave λ s A) ) x s l 1 a s y l 1 and λ y l 1 C A) x s l 1. Fuemoe, we ave λ s A) ) λ a s C A), wic implies a λ ˆϒ s, A) ˆϒ i S, S i, A). And if y x s,en y = max i N w i }. Similaly, we can ge λ c s A) ) λ a s ss s R s A), wic implies a λ ϒ,s A) ϒ i S, S i, A). Case IV: Suppose a w S = y g, w S = x,en y g x g, x y.if y g x,en y g = max i N w i }. Simila o e poof of 5)and4), we ave λ c g A) ) y g l 1 a g x l 1 and λ x l 1 R A) y g l 1.

6 Sang Jounal of Inequaliies and Applicaions 2017) 2017:141 Page 6 of 14 Fuemoe, we ave λ c g A) ) λ a g R A), wic implies a λ ϒ g, A) ϒ i S, S i, A). And if x y g,en x = max i N w i }. Similaly, we can ge λ g A) ) λ a g gg g C g A), wic implies a λ ˆϒ,g A) i S, S ˆϒ i, A). Te poof is compleed. Based on Teoem 1, bounds fo e lages singula value of nonnegaive ecangula ensos ae given. Teoem 2 Le A =a i1 i m ) R [p,q;n,n] +, S be a nonempy pope subse of N, Sbeecomplemen of S in N. Ten L S A) λ 0 U S A), 7) wee L S A)=min ˆL S A), ˆL S A), L S A), L S A) }, U S A)=max Û S A), Û S A), Ũ S A), Ũ S A) } and ˆL S 1 A)= min i A)+[ i A)) 2 +4ai min R A), C A) }] 2 1 }, L S 1 A)= min c i A)+[ c i A)) 2 +4a i min R A), C A) }] 2 1 }, Û S 1 A)= max i A)+[ i A)) 2 +4ai max R A), C A) }] 2 1 }, Ũ S 1 A)= max c i A)+[ c i A)) 2 +4a i max R A), C A) }] 2 1 }. Poof Fis, we pove a e second inequaliy in 7) olds. By Teoem 2 in [2], we know a λ 0 is a singula value of A.Hence,byTeoem1, λ 0 ϒ S A), a is, λ 0 ˆϒ i, A) ϒ i, A) ) o i S, S λ 0 i S, S ˆϒ i, A) ϒ i, A) ). If λ 0 i S, S ˆϒ i, A) ϒ i, A)), en ee ae i S, S suc a λ 0 ˆϒ i, A) oλ 0 ϒ i, A). Wen λ 0 ˆϒ i, A), i.e., λ 0 i A))λ 0 a i maxr A), C A)},ensolvingλ 0

7 Sang Jounal of Inequaliies and Applicaions 2017) 2017:141 Page 7 of 14 gives λ i A)+[ i A)) 2 +4ai max R A), C A) }] 1 2 } 1 max i A)+[ i A)) 2 +4ai max R A), C A) }] 2 1 } = Û S A). Wen λ 0 ϒ i, A), i.e., λ 0 c i A))λ 0 a i maxr A), C A)},ensolvingλ 0 gives λ c i A)+[ c i A)) 2 +4a i max R A), C A) }] 1 2 } 1 max c i A)+[ c i A)) 2 +4a i max R A), C A) }] 2 1 } = Ũ S A). And if λ 0 i S, S ˆϒ i, A) ϒ i, A)), similaly, we can obain a λ 0 Û S A) andλ 0 Ũ S A). Second, we pove a e fis inequaliy in 7) olds. Assume a A is an ieducible nonnegaive ecangula enso, by Teoem 6 of [1], en λ 0 > 0 wi wo posiive lef and ig associaed eigenvecos x =x 1, x 2,...,x n ) T and y =y 1, y 2,...,y n ) T.Le x s = min i S x i}, w i = min i N x i, y i }, x = minx i }, y g = min y i}, i S i S w S = min i S w i}, w S = minw i }. i S y = miny i }, i S We divide e poof ino fou pas. Case I: Suppose a w S = x s, w S = x,eny s x s, y x. i) If x x s,enx s = min i N w i }.Fomes equaliy in 1), we ave λ 0 x l 1 s = a si2 i p 1 q x i2 x ip y 1 y q + a s x p 1 δ i2 ip 1 q =0 a si2 i p 1 q x l 1 s + a s x l 1 δ i2 ip 1 q =0 = s A)xl 1 s + a s x l 1, y q i.e., λ0 s A)) x l 1 s a s x l 1. 8) Moeove, fom e equaliy in 1), we can ge λ 0 x l 1 = i 2,...i p, 1,..., q N a i2 i p 1 q x i2 x ip y 1 y q R A)x l 1 s. 9)

8 Sang Jounal of Inequaliies and Applicaions 2017) 2017:141 Page 8 of 14 Muliplying 8)by9) and noing a x l 1 s x l 1 >0,weave λ0 s A)) λ 0 a s R A). Ten solving fo λ 0 gives λ 0 A) 1 2 s A)+ [ s A)) 2 +4as R A) ] 2 1 } 1 min i A)+[ i A)) 2 +4ai R A) ] 2 1 } ˆL S A). ii) If x s x,enx = min i N w i }. Similaly, we can ge λ 0 A) 1 s 2 A)+ [ s A)) 2 +4as ss s R s A) ] 2 1 } 1 min i i S, S 2 A)+[ i A)) 2 +4ai R A) ] 2 1 } ˆL S A). Case II: Suppose a w S = y g, w S = y,enx g y g, x y. i) If y y g,eny g = min i N w i }.Fomeg equaliy in 2), we ave i.e., λ 0 y l 1 g = a i1 i p g 2 q x i1 x ip y 2 y q + a g x p yq 1 δ i1 ip 2 q =0 a i1 i p g 2 q y l 1 g + a g y l 1 δ i1 ip 2 q =0 = c g A)yl 1 g + a g y l 1, λ0 c g A)) y l 1 g a g y l 1. 10) Moeove, fom e equaliy in 2), we can ge λ 0 y l 1 = i 1,...,i p, 2,..., q N a i1 i p 2 q x i1 x ip y 2 y q C A)y l 1 g. 11) Muliplying 10)by11) and noing a y l 1 g y l 1 >0,weave λ0 c g A)) λ 0 a g C A), wic gives λ 0 1 c 2 g A)+ [ c g A)) 2 +4a g C A) ] 2 1 } 1 min c i A)+[ c i A)) 2 +4a i C A) ] 2 1 } L S A).

9 Sang Jounal of Inequaliies and Applicaions 2017) 2017:141 Page 9 of 14 ii) If y g y,eny = min i N w i }. Similaly, we can ge λ c g A)+[ c g A)) 2 +4ag gg g C g A) ] 1 2 } 1 min c i i S, S 2 A)+[ c i A)) 2 +4a i C A) ] 2 1 } L S A). Case III: Suppose a w S = x s, w S = y,eny s x s, x y.ify x s,enx s = min i N w i }. Simila o e poof of 8)and11), we ave λ0 s A)) x l 1 s a s y l 1 and λ 0 y l 1 C A)x l 1 s. Fuemoe, we ave λ0 s A)) λ 0 a s C A) and λ s A)+ [ s A)) 2 +4as C A) ] 2 1 } 1 min i A)+[ i A)) 2 +4ai C A) ] 2 1 } ˆL S A). And if x s y,eny = min i N w i }. Similaly, we ave λ 0 1 c s 2 A)+ [ c s A)) 2 +4as ss s R s A) ] 2 1 } 1 min c i i S, S 2 A)+[ c i A)) 2 +4a i R A) ] 2 1 } L S A). Case IV: Suppose a w S = y g, w S = x,enx g y g, y x.ifx y g,eny g = min i N w i }. Simila o e poof of 10)and9), we ave λ0 c g A)) y l 1 g a g x l 1 and λ 0 x l 1 R A)y l 1 g.

10 Sang Jounal of Inequaliies and Applicaions 2017) 2017:141 Page 10 of 14 Fuemoe, we ave λ0 c g A)) λ 0 a g R A) and λ 0 1 c 2 g A)+ [ c g A)) 2 +4a g R A) ] 2 1 } 1 min c i A)+[ c i A)) 2 +4a i R A) ] 2 1 } L S A). And if y g x,enx = min i N w i }. Similaly, we ave λ g A)+ [ g A) ) 2 +4ag gg g C g A) ] 1 2 } 1 min i i S, S 2 A)+[ i A)) 2 +4ai C A) ] 2 1 } ˆL S A). Assume a A is a nonnegaive ecangula enso, en by Lemma 3 of [2]andsimila o e poof of Teoem 2 of [2], we can pove a e fis inequaliy in 7) olds.te conclusion follows fom wa we ave poved. Nex, a compaison eoem fo ese bounds in Teoem 2 and Teoem 4 of [2] is given. Teoem 3 Le A =a i1 i m ) R [p,q;n,n] +, S be a nonempy pope subse of N. Ten e bounds in Teoem 2 ae bee an ose in Teoem 4 of [2], a is, min Ri A), C A) } L S A) U S A) max Ri A), C A) }. 1 i, n 1 i, n Poof Hee, only L S A) =minˆl S A), ˆL S A), L S A), L S A)} min 1 i, n R i A), C A)} is poved. Similaly, we can also pove a U S A) max 1 i, n R i A), C A)}. Wiouloss of genealiy, assume a L S A)=ˆL S A), a is, ee ae wo indexes i S, S suc a L S A)= 1 2 i A)+[ i A)) 2 +4ai min R A), C A) }] 1 2 } we can pove i similaly if L S A)=ˆL S A), L S A), L S A), especively). Now, we divide e poof ino wo cases as follows. Case I: Assume a L S A)= 1 2 i A)+[ i A)) 2 +4ai R A) ] 1 2 }. i) If R i A) R A), en a i R A) i A). Wen R A) i A)>0,weave L S A) 1 2 = 1 2 i A)+[ i A)) 2 +4 R A) i A)) R A) ] 1 2 } i A)+[ 2R A) i A)) 2] 1 2 }

11 Sang Jounal of Inequaliies and Applicaions 2017) 2017:141 Page 11 of 14 = 1 2 = R A) i A)+2R A) i A)} min R A) S min Ri A), C A) }. 1 i, n And wen R A) i A) 0, i.e., i A) R A), we ave L S A) 1 i 2 A)+[ i A)) 2] 1 2 } = i A) R A) min R A) S min Ri A), C A) }. 1 i, n ii) If R i A)<R A), en L S A) 1 2 = 1 2 = 1 2 i A)+[ i A)) 2 +4ai R i A) ] 1 2 } i A)+[ i A)) 2 +4ai i A)+a )] 1 2 } i i A)+[ i A)+2a ) 2 ] 1 2 } i = i A)+a i = R i A) min R ia) i S min Ri A), C A) }. 1 i, n Case II: Assume a L S A)= 1 2 i A)+[ i A)) 2 +4ai C A) ] 1 2 }. SimilaoepoofofCaseI,weaveL S A) min 1 i, n R i A), C A)}. Teconclusion follows fom wa we ave poved. 3 Numeical examples In e following, wo numeical examples ae given o veify e eoeical esuls. Example 1 Le A R [2,2;3,3] + wi enies defined as follows: A:,:,1,1)= , A:,:,3,1) 1 1 2, A:,:,2,1)= 4 6 3, A:,:,1,2)= 0 1 0, 1 0 0

12 Sang Jounal of Inequaliies and Applicaions 2017) 2017:141 Page 12 of 14 Figue 1 Te singula value inclusion se ϒ S A) and e exac singula values A:,:,2,2)= 0 2 1, A:,:,1,3)= 2 1 2, A:,:,3,3) A:,:,3,2)= 2 2 2, A:,:,2,3)= 2 3 1, By compuaion, we ge a all diffeen singula values of A ae , , , , 0, , , , , , , , , , , and i) An S-ype singula value inclusion se. Le S = 1}.Obviously, S = 2, 3}.ByTeoem1,eS-ype singula inclusion se is ϒ S A)= z C : z }. Te singula value inclusion se ϒ S A) and e exac singula values ae dawn in Figue 1, wee ϒ S A) is epesened by black solid bounday and e exac singula values ae ploed by ed +. I is easy o see a ϒ S A) can capue all singula values of A fom Figue 1. ii) Te bounds of e lages singula value. By Teoem 4 of [2], we ave 5 λ 0 57.

13 Sang Jounal of Inequaliies and Applicaions 2017) 2017:141 Page 13 of 14 Figue 2 Te singula value inclusion se ϒ S A) and e exac singula values. Le S = 1}, S = 2, 3}.ByTeoem2,weave λ In fac, λ 0 = Tis example sows a e bounds in Teoem 2 ae bee an oseinteoem4of[2]. Example 2 Le A R [2,2;2,2] + wi enies defined as follows: a 1111 = a 1112 = a 1222 = a 2112 = a 2121 = a 2221 =1, oe a ikl = 0. By compuaion, we ge a all diffeen singula values of A ae 0, , 1, 3. i) An S-ype singula value inclusion se. Le S = 1}.Obviously, S = 2, 3}.ByTeoem1,eS-ype singula inclusion se is ϒ S A)= z C : z 3 }. Te singula value inclusion se ϒ S A) and e exac singula values ae dawn in Figue 2, wee ϒ S A) is epesened by black solid bounday and e exac singula values ae ploed by ed +. I is easy o see a ϒ S A) capues exacly all singula values of A fom Figue 2. ii) Te bounds of e lages singula value. By Teoem 2,weave 3 λ 0 3. In fac, λ 0 = 3. Tis example sows a e bounds in Teoem 2 ae sap.

14 Sang Jounal of Inequaliies and Applicaions 2017) 2017:141 Page 14 of 14 4 Conclusions In is pape, we give an S-ype singula value inclusion se ϒ S A) fo ecangula ensos. As an applicaion of is se, an S-ype uppe bound U S A) andans-ype lowe bound L S A) fo e lages singula value λ 0 of a nonnegaive ecangula enso A ae obained andpovedobesapeanosein[2]. Ten, an ineesing poblem is ow o pick S o make ϒ S A) as ig as possible. Bu i is difficul wen e dimension of e enso A is lage. We will coninue o sudy is poblem in e fuue. Acknowledgemens Te auo is vey indebed o e eviewes fo ei valuable commens and coecions, wic impoved e oiginal manuscip of is pape. Tis wok is suppoed by Foundaion of Guizou Science and Tecnology Depamen Gan No. [2015]2073), Naional Naual Science Foundaion of Cina Gan No ) and Naual Science Pogams of Educaion Depamen of Guizou Povince Gan No. [2016]066). Compeing ineess Te auo declaes a ey ave no compeing ineess. Auo s conibuions All auos conibued equally o is wok. All auos ead and appoved e final manuscip. Publise s Noe Spinge Naue emains neual wi egad o uisdicional claims in publised maps and insiuional affiliaions. Received: 18 Apil 2017 Acceped: 7 June 2017 Refeences 1. Cang, KC, Qi, LQ, Zou, GL: Singula values of a eal ecangula enso. J. Ma. Anal. Appl. 370, ) 2. Yang, YN, Yang, QZ: Singula values of nonnegaive ecangula ensos. Fon. Ma. Cina 62), ) 3. Lim, LH: Singula values and eigenvalues of ensos: a vaiaional appoac. In: Poceedings of e IEEE Inenaional Woksop on Compuaional Advances in Muli-Senso Adapive Pocessing CAMSAP 05), pp ) 4. Cang, KC, Peason, K, Zang, T: On eigenvalue poblems of eal symmeic ensos. J. Ma. Anal. Appl. 350, ) 5. Knowles, JK, Senbeg, E: On e ellipiciy of e equaions of non-linea elasosaics fo a special maeial. J. Elas. 5, ) 6. Wang, Y, Aon, M: A efomulaion of e song ellipiciy condiions fo unconsained ypeelasic media. J. Elas. 44, ) 7. Dal, D, Leinass, JM, Myeim, J, Ovum, E: A enso poduc maix appoximaion poblem in quanum pysics. Linea Algeba Appl. 420, ) 8. Einsein, A, Podolsky, B, Rosen, N: Can quanum-mecanical descipion of pysical ealiy be consideed complee? Pys. Rev. 47, ) 9. Li,HB,Huang,TZ,Liu,XP,Li,H:Singulaiy,Wieland slemmaandsingulavalues.j.compu.appl.ma.234, ) 10. Zou, GL, Caccea, L, Qi, LQ: Convegence of an algoim fo e lages singula value of a nonnegaive ecangula enso. Linea Algeba Appl. 438, ) 11. Cen, Z, Lu, LZ: A enso singula values and is symmeic embedding eigenvalues. J. Compu. Appl. Ma. 250, ) 12. Cen, ZM, Qi, LQ, Yang, QZ, Yang, YN: Te soluion meods fo e lages eigenvalue singula value) of nonnegaive ensos and convegence analysis. Linea Algeba Appl. 439, ) 13. Li, CQ, Li, YT, Kong, X: New eigenvalue inclusion ses fo ensos. Nume. Linea Algeba Appl. 21, ) 14. Li, CQ, Jiao, AQ, Li, YT: An S-ype eigenvalue locaion se fo ensos. Linea Algeba Appl. 493, ) 15. He, J, Liu, YM, Ke, H, Tian, JK, Li, X: Bound fo e lages singula value of nonnegaive ecangula ensos. Open Ma. 14, ) 16. Zao,JX,Sang,CL:An S-ype uppe bound fo e lages singula value of nonnegaive ecangula ensos. Open Ma. 14, ) 17. He, J, Liu, YM, Tian, JK, Ren, ZR: New inclusion ses fo singula values. J. Inequal. Appl. 2017, )