Frontiers of Mathematics in China >
Bi-block positive semidefiniteness of bi-block symmetric tensors
Received date: 27 Apr 2020
Accepted date: 06 Nov 2020
Published date: 15 Feb 2021
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The positive definiteness of elasticity tensors plays an important role in the elasticity theory. In this paper, we consider the bi-block symmetric tensors, which contain elasticity tensors as a subclass. First, we define the bi-block M-eigenvalue of a bi-block symmetric tensor, and show that a bi-block symmetric tensor is bi-block positive (semi)definite if and only if its smallest bi-block M-eigenvalue is (nonnegative) positive. Then, we discuss the distribution of bi-block M-eigenvalues, by which we get a sufficient condition for judging bi-block positive (semi)definiteness of the bi-block symmetric tensor involved. Particularly, we show that several classes of bi-block symmetric tensors are bi-block positive definite or bi-block positive semidefinite, including bi-block (strictly) diagonally dominant symmetric tensors and bi-block symmetric (B)B0-tensors. These give easily checkable sufficient conditions for judging bi-block positive (semi)definiteness of a bi-block symmetric tensor. As a byproduct, we also obtain two easily checkable suffcient conditions for the strong ellipticity of elasticity tensors.
Zheng-Hai HUANG , Xia LI , Yong WANG . Bi-block positive semidefiniteness of bi-block symmetric tensors[J]. Frontiers of Mathematics in China, 2021 , 16(1) : 141 -169 . DOI: 10.1007/s11464-021-0874-0
| 1 |
Che H, Chen H, Wang Y. On the M-eigenvalue estimation of fourth-order partially symmetric tensors. J Ind Manag Optim, 2020, 16: 309–324
|
| 2 |
Chirită S, Ghiba I-D. Strong ellipticity and progressive waves in elastic materials with voids. Proc R Soc Lond A, 2010, 466: 439–458
|
| 3 |
Choi M D. Positive semidefinite biquadratic forms. Linear Algebra Appl, 1975, 12: 95–100
|
| 4 |
Ding W, Liu J, Qi L, Yan H. Elasticity M-tensors and the strong ellipticity condition. Appl Math Comput, 2020, 373: 124982
|
| 5 |
Ding W, Luo Z, Qi L.P -tensors, P0-tensors, and their applications. Linear Algebra Appl, 2018, 555: 336–354
|
| 6 |
Ding W, Qi L, Wei Y. M-tensors and nonsingular M-tensors. Linear Algebra Appl, 2013, 439: 3264–3278
|
| 7 |
Ding W, Qi L, Yan H. On some sufficient conditions for strong ellipticity. arXiv: 1705.05081
|
| 8 |
Han D, Dai H H,Qi L. Conditions for strong ellipticity of anisotropic elastic material. J Elasticity, 2009, 97: 1–13
|
| 9 |
Han D, Qi L. A successive approximation method for quantum separability. Front Math China, 2013, 8: 1275–1293
|
| 10 |
Haussühl S. Physical Properties of Crystals: An Introduction.Weinheim: Wiley-VCH Verlag, 2007
|
| 11 |
He J, Li C, Wei Y. M-eigenvalue intervals and checkable sufficient conditions for the strong ellipticity. Appl Math Lett, 2020, 102: 106137
|
| 12 |
Hu S, Huang Z H. Alternating direction method for bi-quadratic programming. J Global Optim, 2011, 51: 429–446
|
| 13 |
Hu S, Huang Z H, Ling C, Qi L. On determinants and eigenvalue theory of tensors. J Symbolic Comput, 2013, 50: 508–531
|
| 14 |
Hu S,Qi L, Song Y, Zhang G. Geometric measure of entanglement of multipartite mixed states. Int J Software Informatics, 2014, 8: 317–326
|
| 15 |
Hu S, Qi L, Zhang G.Computing the geometric measure of entanglement of multipartite pure states by means of non-negative tensors. Phys Rev A, 2016, 93: 012304
|
| 16 |
Huang Z H, Qi L. Positive definiteness of paired symmetric tensors and elasticity tensors. J Comput Appl Math, 2018, 338: 22–43
|
| 17 |
Huang Z H, Qi L. Tensor complementarity problems-part I: basic theory. J Optim Theory Appl, 2019, 183: 1–23
|
| 18 |
Li S, Li C, Li Y. M-eigenvalue inclusion intervals for a fourth-order partially symmetric tensor. J Comput Appl Math, 2019, 356: 391–401
|
| 19 |
Li S H, Li Y T.Bounds for the M-spectral radius of a fourth-order partially symmetric tensor. J Inequal Appl, 2018, 2018: 18, doi.org/10.1186/s13660-018-1610-5
|
| 20 |
Ling C, Nie J, Qi L, Ye Y. Bi-quadratic optimization over unit spheres and semidefinite programming relaxations. SIAM J Optim, 2009, 20: 1286–1310
|
| 21 |
Nie J, Zhang X Z.Positive maps and separable matrices. SIAM J Optim, 2016, 26: 1236–1256
|
| 22 |
Peña J M. A class of P-matrices with applications to the localization of the eigenvalues of a real matrix. SIAM J Matrix Anal Appl, 2001, 22: 1027–1037
|
| 23 |
Peña J M. On an alternative to Gerschgorin circles and ovals of Cassini. Numer Math, 2003, 95: 337–345
|
| 24 |
Qi L. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput, 2005, 40: 1302–1324
|
| 25 |
Qi L, Dai H H, Han D. Conditions for strong ellipticity and M-eigenvalues. Front Math China, 2009, 4: 349–364
|
| 26 |
Qi L, Huang Z H. Tensor complementarity problems-part II: solution methods. J Optim Theory Appl, 2019, 183: 365–385
|
| 27 |
Qi L, Luo Z. Tensor Analysis: Spectral Properties and Special Tensors. Philadelphia: SIAM, 2017
|
| 28 |
Qi L, Song Y. An even order symmetric B tensor is positive definite. Linear Algebra Appl, 2014, 457: 303–312
|
| 29 |
Rosakis P. Ellipticity and deformations with discontinuous deformation gradients in finite elastrostatics. Arch Ration Mech Anal, 1990, 109: 1–37
|
| 30 |
Sfyris D. The strong ellipticity condition under changes in the current and reference configuration. J Elasticity, 2011, 103: 281–287
|
| 31 |
Simpson H C, Spector S J. On copositive matrices and strong ellipticity for isotropic elastic materials. Arch Ration Mech Anal, 1983, 84: 55–68
|
| 32 |
Song Y, Qi L. An initial study on P; P0; B and B0 tensors. arXiv: 1403.1118v3
|
| 33 |
Song Y, Qi L. Properties of some classes of structured tensors. J Optim Theory Appl, 2015, 165: 854–873
|
| 34 |
Straughan B. Stability and uniqueness in double porosity elasticity. Internat J Engrg Sci, 2013, 65: 1–8
|
| 35 |
Wang G, Sun L, Liu L. M-eigenvalues-based sufficient conditions for the positive definiteness of fourth-order partially symmetric tensors. Complexity, 2020, 2020: 2474278
|
| 36 |
Wang Y, Aron M. A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media. J Elasticity, 1996, 44: 89–96
|
| 37 |
Wang Y J, Qi L, Zhang X Z. A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor. Numer Linear Algebra Appl, 2009, 16: 589–601
|
| 38 |
Xu Y, Gu W, Huang Z H. Estimations on upper and lower bounds of solutions to a class of tensor complementarity problems. Front Math China, 2019, 14: 661–671
|
| 39 |
Ye L, Chen Z. Further results on B-tensors with application to location of real eigenvalues. Front Math China, 2017, 12: 1375–1392
|
| 40 |
Yuan P, You L. Some remarks on P; P0; B and B0 tensors. Linear Algebra Appl, 2014, 459: 511–521
|
| 41 |
Zhang L, Qi L, Zhou G. M-tensors and some applications. SIAM J Matrix Anal Appl, 2014, 35: 437–452
|
| 42 |
Zubov L M, Rudev A N. On necessary and sufficient conditions of strong ellipticity of equilibrium equations for certain classes of anisotropic linearly elastic materials. ZAMM Z Angew Math Mech, 2016, 96: 1096–1102
|
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