$l^{p/2,q}$-Singular Values of a Real Partially Symmetric Rectangular Tensor

Abstract

It is known that the strong ellipticity condition of a general non-linearly elastic material can be equivalently transformed into the positive definiteness of a real (p, q)-th order $(m\times n)$-dimensional partially symmetric rectangular tensor $\mathscr {A}$ with p and q even. Furthermore, $\mathscr {A}$ is positive definite if and only if all of its $l^{k,s}$-singular values are positive, where $k,s\geqslant 2$ are even. To determine the positive definiteness of $\mathscr {A}$, we introduce its $l^{p/2,q}$-singular values by letting $k=p/2$ and $s=q$. Subsequently, we adopt two approaches to this problem. The first approach is to construct an interval with parameters that contains all $l^{p/2,q}$-singular values. By optimizing these parameters, we obtain an optimal interval, which yields a criterion for the positive definiteness of $\mathscr {A}$. The second approach is to develop two direct methods for computing all $l^{p/2,q}$-singular values/vectors of $\mathscr {A}$ for the specific cases where $p=4$, $q=2$, or 4, and $m=n=2$. These two methods are based on a case analysis of the $l^{p/2,q}$-singular vectors. Finally, we verify the correctness and effectiveness of the derived interval, criterion, and direct methods via a numerical example.

Keywords

Rectangular tensors / Partially symmetry / $l^{k,s}$-Singular values / $l^{p/2,q}$-Singular values / Positive definiteness / 15A18 / 15A69 / 15A72

References

[1] Chang K, Qi L, Zhou G. Singular values of a real rectangular tensor. J. Math. Anal. Appl., 2010, 370(1): 284-294.

[2] Chen Z, Qi L, Yang Q, Yang Y. The solution methods for the largest eigenvalue (singular value) of nonnegative tensors and convergence analysis. Linear Algebra Appl., 2013, 439(12): 3713-3733.

[3] Cox DA, Little J, O’shea D. Using Algebraic Geometry, 2005, 2, New York, Springer

[4] Dahl, G., Leinaas, J.M., Myrheim, J., Ovrum, E.: A tensor product matrix approximation problem in quantum physics. Linear Algebra Appl. 420(2/3), 711–725 (2007)

[5] Einstein A, Podolsky B, Rosen N. Can quantum-mechanical description of physical reality be considered complete?. Phys. Rev., 1935, 47(10): 777-780.

[6] Gu Y, Wu W. Partially symmetric nonnegative rectangular tensors and copositive rectangular tensors. J. Ind. Manag. Optim., 2019, 15(2): 775-789.

[7] He J, Liu Y, Xu G. Rectangular M-tensors and strong rectangular M-tensors. ScienceAsia, 2021, 47: 257-263.

[8] He J, Liu Y, Xu G, Liu G. V-singular values of rectangular tensors and their applications. J. Inequal. Appl., 2019, 2019: 84.

[9] Huang ZH, Li X, Wang Y. Bi-block positive semidefiniteness of bi-block symmetric tensors. Front. Math. China, 2021, 16(1): 141-169.

[10] Knowles JK, Sternberg E. On the ellipticity of the equations of nonlinear elastostatics for a special material. J. Elast., 1975, 5(3/4): 341-361.

[11] Lim, L.H.: Singular values and eigenvalues of tensors: a variational approach. In: CAMSAP’05: Proceeding of the IEEE International Workshop on Computational Advances in MultiSensor Adaptive Processing, vol. 1, pp. 129–132 (2005)

[12] Ling C, Qi L. $l^{k, s}$-Singular values and spectral radius of rectangular tensors. Front. Math. China, 2013, 8(1): 63-83.

[13] Miao Y, Li C, Wei Y. Z-singular value and Z-singular value inclusion sets for tensors. Jpn. J. Ind. Appl. Math., 2019, 36(3): 1055-1087.

[14] Qi L, Dai HH, Han D. Conditions for strong ellipticity and M-eigenvalues. Front. Math. China, 2009, 4(2): 349-364.

[15] Sang C, Zhao J. Criteria for the strong ellipticity condition of a partially symmetric tensor. Acta Appl. Math., 2022, 180: 10.

[16] Wang C, Wang G, Hou Q. Bounds of the minimum $H^{+}$-singular value and testing the copositiveness for rectangular tensors. J. Ind. Manag. Optim., 2023, 19(9): 6558-6571.

[17] Wang Y, Aron M. A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media. J. Elast., 1996, 44: 89-96.

[18] Yang Q, Li Y. Standard tensor and its applications in problem of singular values of tensors. Front. Math. China, 2019, 14(5): 967-987.

[19] Yang Y, Yang Q. Singular values of nonnegative rectangular tensors. Front. Math. China, 2011, 6(2): 363-378.

[20] Yao H, Long B, Bu C, Zhou J. $l^{k, s}$-Singular values and spectral radius of partially symmetric rectangular tensors. Front. Math. China, 2016, 11(3): 605-622.

[21] Yao, H., Ma, L., Liu, C., Bu, C.: Brualdi-type inclusion sets of $Z$-eigenvalues and $l^{k, s}$-singular values for tensors. Front. Math. China 15(3), 601–612 (2020)

[22] Yao H, Zhang C, Liu L, Zhou J, Bu C. Singular value inclusion sets of rectangular tensors. Linear Algebra Appl., 2019, 576: 181-199.

[23] Zhao J. Two new singular value inclusion sets for rectangular tensors. Linear Multilinear Algebra, 2019, 67(12): 2451-2470.

[24] Zhao J. $l^{p, q}$-Singular values of a partially symmetric rectangular tensor. Pac. J. Optim., 2021, 17(2): 151-173

[25] Zhao J. Conditions of strong ellipticity and calculations of M-eigenvalues for a partially symmetric tensor. Appl. Math. Comput., 2023, 458: 128245

[26] Zhao J. $l^{p/2, q/2}$-Singular values of a real partially symmetric rectangular tensor. Jpn. J. Ind. Appl. Math., 2023, 40(2): 843-875.

[27] Zhao J. $l^{p, q/2}$-Singular values of a real partially symmetric rectangular tensor. J. Ind. Manag. Optim., 2024, 20(3): 967-993.

[28] Zhao J, Li C. Singular value inclusion sets for rectangular tensors. Linear Multilinear Algebra, 2018, 66(7): 1333-1350.

[29] Zhao J, Liu P, Sang C. Shifted inverse power method for computing the smallest M-eigenvalue of a fourth-order partially symmetric tensor. J. Optim. Theory Appl., 2024, 200: 1131-1159.

[30] Zhao J, Luo Y, Sang C. A direct method for calculating M-eigenvalues of an elasticity tensor. Jpn. J. Ind. Appl. Math., 2024, 41: 317-357.

[31] Zhao J, Sang C. Z-eigenvalue intervals of even-order tensors with application to judge the strong ellipticity of an elasticity tensor. Acta Appl. Math., 2022, 182: 5.

[32] Zhou G, Caccetta L, Qi L. Convergence of an algorithm for the largest singular value of a nonnegative rectangular tensor. Linear Algebra Appl., 2013, 438(2): 959-968.

[33] Zubov LM, Rudev AN. On necessary and sufficient conditions of strong ellipticity of equilibrium equations for certain classes of anisotropic linearly elastic materials. Z. Angew. Math. Mech., 2016, 96(9): 1096-1102.

Funding

Guizhou Provincial Science and Technology Projects, China(QKHJC-ZK[2022]YB215)

RIGHTS & PERMISSIONS

Shanghai University