Bounds for eigenvalues of nonsingular H-tensor

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1 Electronc Journal of Lnear Algebra Volume 9 Specal volume for Proceedngs of the Internatonal Conference on Lnear Algebra and ts Applcatons dedcated to Professor Ravndra B. Bapat Artcle 05 Bounds for egenvalues of nonsngular H-tensor Xue-Zhong Wang School of Mathematcs and Statstcs, Hex Unversty, xuezhong-wang@6.com Ymn We School of Mathematcs and Computer Scence, Guzhou Normal Unversty, ymwe_cn@yahoo.com Follow ths and addtonal works at: Recommended Ctaton Wang, Xue-Zhong and We, Ymn. (05), "Bounds for egenvalues of nonsngular H-tensor", Electronc Journal of Lnear Algebra, Volume 9, pp DOI: Ths Artcle s brought to you for free and open access by Wyomng Scholars Repostory. It has been accepted for ncluson n Electronc Journal of Lnear Algebra by an authorzed edtor of Wyomng Scholars Repostory. For more nformaton, please contact scholcom@uwyo.edu.

2 Electronc Journal of Lnear Algebra ISSN Volume 9, pp. 3-6, September 05 BOUNDS FOR EIGENVALUES OF NONSINGULAR H-TENSOR XUE-ZHONG WANG AND YIMIN WEI Abstract. The bounds for the Z-spectral radus of nonsngular H-tensor, the upper and lower bounds for the mnmum H-egenvalue of nonsngular (strong) M-tensor are studed n ths paper. Sharper bounds than known bounds are obtaned. Numercal examples llustrate that our bounds gve tghter bounds. Dedcated to Professor Ravndra B. Bapat on the occason of hs 60th brthday Key words. M-tensor, H-tensor, Z-spectral radus, Mnmum H-egenvalue. AMS subject classfcatons. 5A8, 5A69, 65F5, 65F0.. Introducton. Egenvalue problems of hgher order tensors have become an mportant topc n appled mathematcs branch, numercal multlnear algebra, and t has a wde range of practcal applcatons [, 3, 4,, 8,, 3, 4, 5, 6, 9, 0, ]. A tensor can be regarded as a hgher-order generalzaton of a matrx. Let C (respectvely, R) be the complex (respectvely, real) feld. An m-order n-dmensonal square tensor A wth n m entres can be defned as follows, A = (a... m ), a... m C,,,..., m n. Let A be an m-order n-dmensonal tensor, and x C n. Then (.) Ax m = n,,..., m= a... m x x...x m, and Ax m s a vector n C n, wth ts th component defned by (Ax m ) = n, 3,..., m= a... m x x 3 x m. Receved by the edtorson February 5, 05. Accepted forpublcaton onjuly 0, 05. Handlng Edtor: Manjunatha Prasad Karantha. School of Mathematcs and Statstcs, Hex Unversty, Zhangye, , P. R. Chna (xuezhongwang77@6.com ). Ths author s supported by the Natonal Natural Scence Foundaton of Chna under grant 737. School of Mathematcs and Computer Scence, Guzhou Normal Unversty, Guyang 55000, P.R. Chna (ymwe cn@yahoo.com). Ths author s supported by the Natonal Natural Scence Foundaton of Chna under grant

3 Electronc Journal of Lnear Algebra ISSN Volume 9, pp. 3-6, September 05 4 X.Z. Wang, and Y. We Let r be a postve nteger. Then x [r] = [x r,xr,...,xr n ] s a vector n C n, wth ts th component defned by x r. The followng two defntons were frst ntroduced and studed by Q and Lm, respectvely. Defnton.. ([8,, 4]) Let A be an m-order n-dmensonal real tensor. A par (λ,x) C (C n \0) s called an egenvalue-egenvector (or smply egenpar) of A, f t satsfes the equaton Ax m = λx [m ]. We call (λ,x) an H-egenpar, f both λ and x are real. Defnton.. ([8,, 4]) Let A be an m-order n-dmensonal real tensor. A par (λ,x) C (C n \0) s called an E-egenvalue and E-egenvector (or smply E-egenpar) of A, f they satsfy the equaton Ax m = λx, x x =. We call (λ,x) a Z-egenpar, f both λ and x real. Here x denotes the transpose of x. In [8], He and Huang presented the defnton of the Z-spectral radus of A as follows. Defnton.3. ([, 8]) Suppose that A s an m-order n-dmensonal real tensor. Let σ(a) denote the Z-spectrum of A by the set of all Z-egenvalues of A. Assume that σ(a). Then the Z-spectral radus of A s denoted by ρ(a) = sup λ : λ σ(a). Partcularly, f A s an m-order n-dmensonal nonnegatve tensor, then ρ(a) = max λ : λ σ(a). Recently, many contrbutons have been made on the bounds of the spectral radus of nonnegatve tensor n [, 0, 3, 4]. Smlarly, bounds for the Z-spectral radus were gven n [8] for the H-tensors. Also, n [7], He and Huang obtaned the upper and lower bounds for the mnmum H-egenvalue of nonsngular (strong) M-tensors. In ths paper, our purpose s to propose sharper bounds for the Z-spectral radus of nonsngular H-tensors and for the mnmum H-egenvalue of nonsngular (strong) M-tensors.

4 Electronc Journal of Lnear Algebra ISSN Volume 9, pp. 3-6, September 05 Bounds for Egenvalues of Nonsngular H-tensor 5. Prelmnares. We start ths secton wth some fundamental notons and propertes on tensors. An m-order n-dmensonal tensor A s called nonnegatve ([, 3,9,6, 0,]), feachentrysnonnegatve. SmlartoZ-matrces,wedenotetensors wth all non-postve off-dagonal entres by Z-tensors. The m-order n-dmensonal dentty tensor, denoted by I = (δ... m ), s the tensor wth entres, = δ... m = = = m, 0, otherwse. The tensor D = (d... m ) s the dagonal tensor of A = (a... m ), f d... m = a... m, = = = m, 0, otherwse. Defnton.. ([8]) Let A and B be two m-order n-dmensonal tensors. If there exsts matrces P and Q of n-order wth PIQ = I such that B = PAQ, then we say that the two tensors are smlar. Let the tensor F be assocated wth an undrected d-partte graph G(F) = (V,E(F)), the vertex set of whch s the dsjont unon V = d j= V j, wth V j = [m j ],j [d]. The edge ( k, l ) V k V l,k l belongs to E(F) f and only f f,,..., d > 0 for some d ndces,..., d \ k, l. The tensor F s called weakly rreducble f the graph G(F) s connected. We call F rreducble f for each proper nonempty subset I V, the followng condton holds: let J := V \I. Then there exsts k [d], k I V k and j J V j for each j [d]\k such that f,..., d > 0. Ths defnton of rreducblty agrees wth [, 3]. Fredland et al. [6] showed that f F s rreducble then F s weakly rreducble and presented the followng results. Lemma.. ([6]) If the nonnegatve tensor A s rreducble, then A s weakly rreducble. For m =, A s rreducble f and only f A s weakly rreducble. Lemma. llustrates that a nonnegatve rreducble tensor must be weakly rreducble. For a general tensor A = (a... m ), a... m C, we can draw the followng concluson. Lemma.3. If a tensor A s rreducble, then A s weakly rreducble. For m =, A s rreducble f and only f A s weakly rreducble. Proof. Let A = D E, where D s the dagonal tensor of A. If A s rreducble, t s equvalent that E s rreducble. Note that E s a nonnegatve tensor, by Lemma., E s weakly rreducble, and then A s weakly rreducble. Smlar to the proof of [6], we can get case m =.

5 Electronc Journal of Lnear Algebra ISSN Volume 9, pp. 3-6, September 05 6 X.Z. Wang, and Y. We Lemma.4. ([4]) The product of the egenvalues λ of tensor A s equal to det(a), that s, det(a) = n(m ) n = We call tensor A s nonsngular, f det(a) 0. Defnton.5. ([5, 3]) We call a tensor A an M-tensor, f there exst a nonnegatve tensor B and a postve real number η ρ(b) such that A = ηi B. If η > ρ(b) then A s called a nonsngular (strong) M-tensor. In [3], Zhang et al. obtaned the followng result for the H-egenvalues of a nonsngular (strong) M-tensor. Lemma.6. ([3]) Let A be a nonsngular (strong) M-tensor and τ(a) denote the mnmal value of the real part of all egenvalues of A. Then τ(a) > 0 s an H- egenvalue of A wth a nonnegatve egenvector. If A s weakly rreducble Z-tensor, then τ(a) > 0 s the unque egenvalue wth a postve egenvector. Yang and Yang [0], Yuan and You [] showed that f λ. (.) B = D (m ) AD (m ), where D s a dagonal nonsngular matrx, then A and B are smlar. It s easy to see that the smlarty relaton s an equvalent relaton, and smlar tensors have the same characterstc polynomals, and thus they have the same spectrum (as a mult-set). Now, we ntroduce the comparson tensor of any tensor A. Defnton.7. ([5]) Let A = (a... m ) be an m-order and n-dmensonal tensor. We call a tensor M(A) = (m... m ) the comparson tensor of A f m... m = a... m, (... m ) = (... ), a... m, (... m ) (... ). In the followng, some basc defntons are gven, whch wll be used n the subsequent dscusson. In [5], Dng et al. extended H-matrces to H-tensors as follows. Defnton.8. ([5]) We call a tensor A an H-tensor, f ts comparson tensor s an M-tensor; we call t as a nonsngular H-tensor, f ts comparson tensor s a nonsngular M-tensor.

6 Electronc Journal of Lnear Algebra ISSN Volume 9, pp. 3-6, September 05 Bounds for Egenvalues of Nonsngular H-tensor 7 Very recently, Kannan et al. [7] establshed some propertes of strong H-tensors and general H-tensors. Remark.9. It follows from defnton.8 that an M-tensor s an H-tensor and a nonsngular M-tensor s a nonsngular H-tensor. Defnton.0. ([5]) Let A be an m-order and n-dmensonal tensor. A s quas-dagonally domnant, f there exsts a postve vector x = (x,x,...,x n ) such that (.) a... x m ( 3... m) (...) a... m x x 3...x m, =,,...,n. If the strct nequalty holds n (.) for all, A s called quas-strctly dagonally domnant. Lemma.. ([5]) A tensor A s a nonsngular H-tensor f and only f t s quas-strctly dagonally domnant. 3. Bounds for the spectral radus of H-tensors. In ths secton, we present some bounds for the Z-spectral radus of H-tensors. For convenence, let N =,,...,n. We denote by R (A) and R(A) the sum of the th row and the maxmal row sum of A, respectvely,.e., R (A) = n, 3,..., m= a... m, R(A) = maxr (A). In [], Chang, Pearson, and Zhang have gven the followng bounds for the Z- egenvalues of an m-order n-dmensonal tensor A. Lemma 3.. ([]) Let A be an m-order and n-dmensonal tensor wth σ(a). Then ρ(a) n nmax a... m = nr(a). N, 3,..., m= For postvely homogeneous operators, Song and Q [9] establshed the relatonshp between the Gelfand formula and the spectral radus, as well as the upper bound of the spectral radus. Followng the Corollary 4.5 n [9], He and Huang [8] presented the followng lemma. Lemma 3.. ([8, 9]) Let A be an m-order and n-dmensonal tensor wth σ(a). Then n ρ(a) max a... m = R(A). N, 3,..., m=

7 Electronc Journal of Lnear Algebra ISSN Volume 9, pp. 3-6, September 05 8 X.Z. Wang, and Y. We Based on the above lemma, we obtan some upper bounds for the Z-spectral radus when A s a nonsngular H-tensor as follows. Theorem 3.3. Let A be an m-order and n-dmensonal nonsngular H-tensor wth σ(a). Then ρ(a) max N a.... Proof. Snce A s a nonsngular H-tensor, there exsts a postve dagonal matrx X =dag(x,x,...,x n ) such that AX (m ) s strctly dagonally domnant. Then s also strctly dagonally domnant,.e., a... > (, 3,...,m) (,,...,) X (m ) AX (m ), a... m x x 3...x m x m, N. Because X (m ) AX (m ) and A are smlar, t follows that ρ(a) = ρ(x (m ) AX (m ) ) R(X (m ) AX (m ) ) n = max, 3,..., m= = max( a... + < max a.... a...m x x 3...xm x m (, 3,...,m) (,,...,) a...m x x 3...xm ) x m By the above theorem, the followng corollary can be obtaned easly. Corollary 3.4. If A s an m-order and n-dmensonal nonsngular H-tensor wth σ(a), then ρ(a) mn R(A),max a.... N Corollary 3.5. If A s an m-order and n-dmensonal nonsngular M-tensor wth σ(a), then ρ(a) mn R(A),max a.... N

8 Electronc Journal of Lnear Algebra ISSN Volume 9, pp. 3-6, September 05 Bounds for Egenvalues of Nonsngular H-tensor 9 Remark 3.6. In fact, the bound of Theorem 3.3 s not better than the bound n Lemma 3. for dagonally domnant H-tensors. However, by Lemma. we know that H-tensors are not necessary dagonally domnant. Thus, the bound gven n Theorem 3.3 s sharper than the one gven n Lemma 3. for non-dagonally domnant H-tensors. The followng example llustrates the same. Example 3.7. Let A = (a jk ) be an 3-order -dmenson tensor wth the form, a =., a =, a =, a =, a =, a =, a =, a =.. It s easy to check that A s quas-strctly dagonally domnant and then A s an nonsngular H-tensor. By Lemma 3., we have, ρ(a) nr(a) = By Lemma 3., we obtan the upper bound, ρ(a) R(A) = 4.. Now from Theorem 3.3, we have the followng bound: ρ(a) max a =.. Obvously, the bound gven n Theorem 3.3 s sharper than those gven n Lemma 3. and Lemma Bounds for the mnmum egenvalue of M-tensors. In ths secton, we consder the mnmum H-egenvalue of M-tensors. We adopt the followng notaton throughout ths secton. We defne a nonnegatve matrx M(A), where (M(A)) j = r (A), = j, a j...j, j. r j (A) = δ...m =0 δ j...m =0 r (A) a j...j, and j (A) = [a... a jj...j +r j (A)] 4a j...j r j (A), wth r (M(A)) = M(A) j, r (A) = r (A) r (M(A)),

9 Electronc Journal of Lnear Algebra ISSN Volume 9, pp. 3-6, September 05 0 X.Z. Wang, and Y. We and j (A) = [a... a jj...j + r (A)] +4r (M(A))r j (A). Lemma 4.. Let A be a weakly rreducble M-tensor and t = k,j a k...k, N. () If 0 t [a... a jj...j +r j (A) r j(a)],,j N, then j (A) j (A). () If t [a... a jj...j +r j (A) r j(a)],,j N, then j (A) j (A). Proof. For convenence, denote a = a... a jj...j + r j (A), notce that r (A) = r j (A) t. Thus j (A) j (A) = a 4a j...j r j (A) (a t ) 4[(t a j...j )r j (A)] = t +[a r j(a)]t. The equaton t +[a r j(a)]t = 0 has two roots t = 0 and t = [a r j (A)]. Therefore, f 0 t [a r j (A)]. Thus j (A) j (A), and f t [a r j (A)], then j (A) j (A). In [7], He and Huang gave the followng bounds for the mnmum H-egenvalue of rreducble M-tensors. Lemma 4.. ([7]) Let A be an rreducble M-tensor. Then τ(a) mn N a... Lemma 4.3. ([7]) Let A = (a... m ) be an rreducble M-tensor. Then (4.) mn max a... +a jj...j r j (A) j (A) τ(a) a... +a jj...j r j (A) j (A). For the weakly rreducble M-tensor, we have a result smlar to that of Lemma 4. n the followng. Lemma 4.4. Let A be a weakly rreducble M-tensor. Then τ(a) mn N a... Proof. The proof s smlar to that of Theorem. n [7], and omt t. Based on the above lemma, we derve the bounds for the mnmum H-egenvalue of weakly rreducble M-tensors as follows.

10 Electronc Journal of Lnear Algebra ISSN Volume 9, pp. 3-6, September 05 Bounds for Egenvalues of Nonsngular H-tensor Theorem 4.5. Let A = (a... m ) be a weakly rreducble M-tensor. Then mn a... +a jj...j r (A) j (A) τ(a) (4.) a... +a jj...j r (A) j (A). max Proof. Let x > 0 be an egenvector of A correspondng to τ(a)..e., (4.3) Ax m = τ(a)x [m ]. Suppose that From (4.3), we have [τ(a) a tt...t ]x m t = x t x s max N x : t, s. δ...m =0 ( 3... m) (jj...j) a t... m x x 3...x m + j t a tj...j x m j. Takng modulus n the above equaton and usng the trangle nequalty gves, τ(a) a tt...t x m t a t... m x x 3...x m + a tj...j x m j δ...m =0 j t ( 3... m) (jj...j) a t... m x m t + a tj...j x m s j t Note that τ(a) a tt...t, and δ...m =0 ( 3... m) (jj...j) = r t (A)x m t +r t (M)x m s. Equvalently [a tt...t τ(a)]x m t r t (A)x m t (4.4) [a tt...t τ(a) r t (A)]x m t From (4.3), we also obtan (4.5) [a ss...s τ(a)]x m s Multplyng nequaltes (4.4) wth (4.5), we have +r t (M)x m s. r t (M)x m s. r s (A)x m t. [a tt...t τ(a) r t (A)][a ss...s τ(a)]x m t x m s r t (M)r s (A)x m s x m t.

11 Electronc Journal of Lnear Algebra ISSN Volume 9, pp. 3-6, September 05 X.Z. Wang, and Y. We Note that x m t x m s < 0, and (4.6) [a tt...t τ(a) r t (A)][a ss...s τ(a)] r t (M)r s (A). Ths s τ(a) [a tt...t +a ss...s r t (A)]τ(A) r t (M)r s (A)+[a tt...t r t (A)]a ss...s 0. Note that [a tt...t +a ss...s r t (A)] 4[a ss...s r t (A)]a tt...t = [a tt...t a ss...s + r t (A)]. Ths gves the followng bound for τ(a), On the other hand, let From (4.3), we have τ(a) a tt...t +a ss...s r t (A) ts (A) mn (4.7) (a uu...u τ(a))x m u = a... +a jj...j r (A) j (A). x l x u mn N x : t, s. l. δ u...m =0 a u... m x x 3...x m r u (A)x m and (a ll...l τ(a))x m l Then = δ l...m =0 ( 3... m) (jj...j) r l (A)x m l a l... m x x 3...x m a lj...j x m j j l +r l (M)x m u. (4.8) [a ll...l τ(a) r l (A)]x m l Multplyng nequaltes (4.7) wth (4.8), we have r l (M)x m u. (4.9) [a uu...u τ(a)][a ll...l τ(a) r l (A)] r l (M)r u (A). Inequalty (4.9) s equvalent to τ(a) [a ll...l +a uu...u r l (A)]τ(A) r l (M)r u (A)+[a ll...l r l (A)]a uu...u 0. Ths gves the followng bound for τ(a), τ(a) a ll...l +a uu...u r l (A) lu (A) a... +a jj...j r j (A) j (A) max.

12 Electronc Journal of Lnear Algebra ISSN Volume 9, pp. 3-6, September 05 Ths completes the proof. Bounds for Egenvalues of Nonsngular H-tensor 3 In what follows, we wll show the bounds n Theorem 4.5 are tghter and sharper than those of Lemma 4.3. Theorem 4.6. Under the condtons of Lemma 4.. If [ ] 0 t a... a jj...j +r j (A) r j(a),,j N, then mn mn a... +a jj...j r j (A) j (A) a... +a jj...j r (A) j (A). Proof. From the Lemma 4., f 0 t [a... a jj...j +r j (A) r j(a), then j (A) j (A). Note that r (A) = r j (A) t, and then a... +a jj...j r j (A) j (A) a... +a jj...j r (A) j (A),,j N, whch mples that mn mn a... +a jj...j r j (A) j (A) a... +a jj...j r (A) j (A). Remark 4.7. From Theorem 4.6, we can see that the lower bound of τ(a) n Theorem 4.5 s sharper than those of Lemma 4.3, f [ ] 0 t a... a jj...j +r j (A) r j(a),,j N. Theorem 4.8. Under the condtons of Lemma 4.. If [ ] t a... a jj...j +r j (A) r j(a) +,,j N, then max max a... +a jj...j r (A) j (A) a... +a jj...j r j (A) j (A).

13 Electronc Journal of Lnear Algebra ISSN Volume 9, pp. 3-6, September 05 4 X.Z. Wang, and Y. We Proof. From the proof of Lemma 4., we know that (4.0) j (A) j (A)+t = t +[(a r j (A))+]t. Because equaton (4.0) has two roots t = 0 and t = [a r j (A)]+. Therefore, f t (a r j (A))+, then Note that r (A) = r j (A) t, we have j (A) j (A) t. r (A)+ j (A) r j (A)+ j (A). Hence max max a... +a jj...j r (A) j (A) a... +a jj...j r j (A) j (A). Remark 4.9. From Theorem 4.8, we can see that the upper bound of τ(a) n Theorem 4.5 s sharper than those n Lemma 4.3, f t [a... a jj...j + r j (A) r j (A)]+,,j N. Remark 4.0. Snce t 0, f 0 t [a... a jj...j + r j (A) r j(a)] for some, and t [a... a jj...j +r j (A) r j(a)]+ for some other, we can see that the upper and lower bounds of τ(a) n Theorem 4.5 are tghter than those of Lemma 4.3. The followng example shows ths. Example 4.. Let A = (a jk ) be an 4-order 3-dmenson tensor wth the form, a = a = 5, a 333 = a 444 = 4, a jj =, j, a = 0.5, a =, a jk = 0, otherwse. By Lemma 4.3, we have the bound 0.64 τ(a) We have our new bounds from Theorem τ(a).769.

14 Electronc Journal of Lnear Algebra ISSN Volume 9, pp. 3-6, September 05 Bounds for Egenvalues of Nonsngular H-tensor 5 5. Concluson. In ths paper, the Z-spectral radus for nonsngular H-tensor and the mnmum H-egenvalue of nonsngular (strong) M-tensor are studed. Furthermore, we prove that the results of ths paper are sharper than those of [, 8] and [7]. Acknowledgement. The authors would lke to thank the Edtor, Prof. Manjunatha Prasad Karantha and the referee for ther valuable and detaled comments on our manuscrpt. We also thank Prof. L. Q for remndng us of the recent paper [7] durng our revson. REFERENCES [] K. C. Chang, K. J. Pearson, and Tan Zhang. Some varatonal prncples for Z-egenvalues of nonnegatve tensors. Lnear Algebra Appl., 438():466 48, 03. [] K. C. Chang, Kelly Pearson, and Tan Zhang. Perron-Frobenus theorem for nonnegatve tensors. Commun. Math. Sc., 6():507 50, 008. [3] K. C. Chang, Kelly Pearson, and Tan Zhang. On egenvalue problems of real symmetrc tensors. J. Math. Anal. Appl., 350():46 4, 009. [4] K. C. Chang, Kelly J. Pearson, and Tan Zhang. Prmtvty, the convergence of the NQZ method, and the largest egenvalue for nonnegatve tensors. SIAM J. Matrx Anal. Appl., 3(3):806 89, 0. [5] Weyang Dng, Lqun Q, and Ymn We. M-tensors and nonsngular M-tensors. Lnear Algebra Appl., 439(0): , 03. [6] S. Fredland, S. Gaubert, and L. Han. Perron-Frobenus theorem for nonnegatve multlnear forms and extensons. Lnear Algebra Appl., 438(): , 03. [7] Jun He and Tng-Zhu Huang. Inequaltes for M-tensors. Journal of Inequaltes and Applcatons, 04():4, 04. [8] Jun He and Tng-Zhu Huang. Upper bound for the largest Z-egenvalue of postve tensors. Appl. Math. Lett., 38:0 4, 04. [9] ShengLong Hu, ZhengHa Huang, and LQun Q. Strctly nonnegatve tensors and nonnegatve tensor partton. Sc. Chna Math., 57():8 95, 04. [0] Chaoqan L, Yaotang L, and Xu Kong. New egenvalue ncluson sets for tensors. Numercal Lnear Algebra wth Applcatons, ():39 50, 04. [] LH Lm. Sngular values and egenvalues of tensors: A varatonal approach. In IEEE CAMSAP 005: Frst Internatonal Workshop on Computatonal Advances n Mult-Sensor Adaptve Processng, pages 9 3. IEEE, 005. [] Yongjun Lu, Guanglu Zhou, and Nur Fadhlah Ibrahm. An always convergent algorthm for the largest egenvalue of an rreducble nonnegatve tensor. J. Comput. Appl. Math., 35():86 9, 00. [3] Mchael Ng, Lqun Q, and Guanglu Zhou. Fndng the largest egenvalue of a nonnegatve tensor. SIAM J. Matrx Anal. Appl., 3(3): , 009. [4] Lqun Q. Egenvalues of a real supersymmetrc tensor. J. Symbolc Comput., 40(6):30 34, 005. [5] Lqun Q. Egenvalues and nvarants of tensors. J. Math. Anal. Appl., 35(): , 007. [6] Lqun Q. Symmetrc nonnegatve tensors and copostve tensors. Lnear Algebra Appl., 439():8 38, 03. [7] M. Rajesh Kanan, N. Shaked-Monderer, and A. Berman. Some propertes of strong H-tensors and general H-tensors. Lnear Algebra Appl., 476:4 55, 05.

15 Electronc Journal of Lnear Algebra ISSN Volume 9, pp. 3-6, September 05 6 X.Z. Wang, and Y. We [8] Ja-Yu Shao. A general product of tensors wth applcatons. Lnear Algebra Appl., 439(8): , 03. [9] Ysheng Song and Lqun Q. Spectral propertes of postvely homogeneous operators nduced by hgher order tensors. SIAM J. Matrx Anal. Appl., 34(4):58 595, 03. [0] Qngzh Yang and Yunng Yang. Further results for Perron-Frobenus theorem for nonnegatve tensors II. SIAM J. Matrx Anal. Appl., 3(4):36 50, 0. [] Yunng Yang and Qngzh Yang. Further results for Perron-Frobenus theorem for nonnegatve tensors. SIAM J. Matrx Anal. Appl., 3(5):57 530, 00. [] Pngzh Yuan and Lhua You. On the smlarty of tensors. Lnear Algebra Appl., 458:534 54, 04. [3] Lpng Zhang, Lqun Q, and Guanglu Zhou. M-tensors and some applcatons. SIAM J. Matrx Anal. Appl., 35():437 45, 04.