TM-Eigenvalues of Odd-Order Tensors

doi:10.1007/s42967-021-00172-z

Communications on Applied Mathematics and Computation ›› 2022, Vol. 4 ›› Issue (4): 1258-1279.doi: 10.1007/s42967-021-00172-z

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T_M-Eigenvalues of Odd-Order Tensors

M. Pakmanesh, Hamidreza Afshin

Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran

收稿日期:2021-05-07 修回日期:2021-08-26 出版日期:2022-12-20 发布日期:2022-09-26
通讯作者: Hamidreza Afshin,E-mail:afshin@vru.ac.ir;M. Pakmanesh,E-mail:mehri.pakmanesh@stu.vru.ac.ir E-mail:afshin@vru.ac.ir;mehri.pakmanesh@stu.vru.ac.ir

T_M-Eigenvalues of Odd-Order Tensors

M. Pakmanesh, Hamidreza Afshin

Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran

Received:2021-05-07 Revised:2021-08-26 Online:2022-12-20 Published:2022-09-26

摘要/Abstract

摘要： In this paper, we propose a definition for eigenvalues of odd-order tensors based on some operators. Also, we define the Schur form and the Jordan canonical form of such tensors, and discuss commuting families of tensors. Furthermore, we prove some eigenvalue inequalities for Hermitian tensors. Finally, we introduce characteristic polynomials of odd-order tensors.

关键词: T_M-product, T_M-eigenvalue, T_M-Schur form, T_M-Jordan canonical form, Oddorder tensor, F_M-upper (lower) triangular tensor

Abstract: In this paper, we propose a definition for eigenvalues of odd-order tensors based on some operators. Also, we define the Schur form and the Jordan canonical form of such tensors, and discuss commuting families of tensors. Furthermore, we prove some eigenvalue inequalities for Hermitian tensors. Finally, we introduce characteristic polynomials of odd-order tensors.

Key words: T_M-product, T_M-eigenvalue, T_M-Schur form, T_M-Jordan canonical form, Oddorder tensor, F_M-upper (lower) triangular tensor

中图分类号:

M. Pakmanesh, Hamidreza Afshin. T_M-Eigenvalues of Odd-Order Tensors[J]. Communications on Applied Mathematics and Computation, 2022, 4(4): 1258-1279.

参考文献

1. Chan, T., Yang, T.:Polar n-complex and n-bicomplex singular value decomposition and principal component pursuit. IEEE Trans. Signal Process. 64, 6533-6544 (2016)
2. Golub, G.H., Van Loan, C.F.:Matrix Computations, 4th edn. Johns Hopkins University Press, Baltimore (2013)
3. Hao, N., Kilmer, M., Braman, K., Hoover, R.:Facial recognition using tensor-tensor decompositions. SIAM J. Imaging Sci. 6, 437-463 (2013)
4. Higham, N.J.:Functions of Matrices:Theory and Computation. SIAM, Philadelphia (2008)
5. Horn, A.R., Johnson, C.R.:Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)
6. Hu, W., Yang, Y., Zhang, W., Xie, Y.:Moving object detection using tensor-based low-rank and saliently fused-sparse decomposition. IEEE Trans. Image Process. 26, 724-737 (2017)
7. Kilmer, M., Braman, K., Hao, N., Hoover, R.:Third-order tensors as operators on matrices:a theo-retical and computational framework with applications in imaging. SIAM J. Matrix Anal. Appl. 34, 148-172 (2013)
8. Kilmer, M., Martin, C.:Factorization strategies for third-order tensors. Linear Algebra Appl. 435, 641-658 (2011)
9. Kong, H., Xie, X., Lin, Z.:t-Schatten-p norm for low-rank tensor recovery. IEEE J. Sel. Top. Signal Process. 12, 1405-1419 (2018)
10. Liu, T., Chen, L., Zhu, C.:Improved robust tensor principal component analysis via low-rank core matrix. IEEE J. Sel. Top. Signal Process. 12, 1378-1389 (2018)
11. Liu, W., Jin, X.:A study on T-eigenvalues of third-order tensors. Linear Algebra Appl. 612, 357-374 (2021)
12. Long, Z., Liu, Y., Chen, L., Zhu, C.:Low rank tensor completion for multiway visual data. Signal Process. 155, 301-316 (2019)
13. Lund, K.:The tensor t-function:a definition for functions of third-order tensors. Numer Linear Algebra Appl. 27, e2288 (2020)
14. Martin, C., Shafer, R., Larue, B.:An order-p tensor factorization with applications in imaging. SIAM J. Sci. Comput. 35, A474-A490 (2013)
15. Miao, Y., Qi, L., Wei, Y.:T-Jordan canonical form and T-Drazin inverse based on the T-product. Communications on Applied Mathematics and Computation, 3(2), 201-220 (2021)
16. Miao, Y., Qi, L., Wei, Y.:Generalized tensor function via the tensor singular value decomposition based on the T-product. Linear Algebra Appl. 590, 258-303 (2020)
17. Soltani, S., Kilmer, M., Hansen, P.:A tensor-based dictionary learning approach to tomographic image reconstruction. BIT Numer. Math. 56, 1425-1454 (2016)
18. Tarzanagh, D., Michailidis, G.:Fast randomized algorithms for t-product based tensor operations and decompositions with applications to imaging data. SIAM J. Imaging Sci. 11, 2629-2664 (2018)
19. Wang, A., Lai, Z., Jin, Z.:Noisy low-tubal-rank tensor completion. Neurocomputing 330, 267-279 (2019)

[1]	Yun Miao, Liqun Qi, Yimin Wei. T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product[J]. Communications on Applied Mathematics and Computation, 2021, 3(2): 201-220.
[2]	Guangbin Wang, Fuping Tan. Some Criteria for H-Tensors[J]. Communications on Applied Mathematics and Computation, 2020, 2(4): 641-651.

T_M-Eigenvalues of Odd-Order Tensors

T_M-Eigenvalues of Odd-Order Tensors

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