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Generalized eigenvalue for even order tensors via Einstein product and its applications in multilinear control systems

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Abstract

This paper devotes to the generalized eigenvalues for even order tensors. We extend classical spectral theory for matrix pairs to the multilinear case, including the generalized Schur decomposition, the Geršgorin circle theorem, and the Bauer–Fike theorem for regular tensor pairs of even order. We introduce the backward errors and \(\epsilon \)-pseudospectrums for generalized tensor eigenvalues in normwise and componentwise, respectively, and particularize the application in stability analysis for the generalized multilinear systems. By the normwise pseudospectral theory, we obtain a lower bound for the distance from a regular tensor pair to singularity, and a formulation of the distance from a reachable multilinear time invariant control system to unreachability is given.

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References

  • Bai Z, Demmel J, Dongarra J, Ruhe A, Van der Vorst H (2000) Templates for the solution of algebraic eigenvalue problems: a practical guide. Society for Industrial and Applied Mathematics, Philadelphia

    MATH  Google Scholar 

  • Bauer FL, Fike CT (1960) Norms and exclusion theorems. Numer Math 2(1):137–141

    MathSciNet  MATH  Google Scholar 

  • Brazell M, Li N, Navasca C, Tamon C (2013) Solving multilinear systems via tensor inversion. SIAM J Matrix Anal Appl 34(2):542–570

    MathSciNet  MATH  Google Scholar 

  • Burke JV, Lewis AS, Overton ML (2004) Pseudospectral components and the distance to uncontrollability. SIAM J Matrix Anal Appl 26(2):350–361

    MathSciNet  MATH  Google Scholar 

  • Byers R, He C, Mehrmann V (1998) Where is the nearest non-regular pencil? Linear Algebra Appl 285(1–3):81–105

    MathSciNet  MATH  Google Scholar 

  • Cao Z, Xie P (2022) On some tensor inequalities based on the T-product. Linear Multilinear Algebra. https://doi.org/10.1080/03081087.2022.2032567

    Article  Google Scholar 

  • Cardoso J-F (1999) High-order contrasts for independent component analysis. Neural Comput 11(1):157–192

    Google Scholar 

  • Chandra Rout N, Panigrahy K, Mishra D (2022) A note on numerical ranges of tensors. Linear Multilinear Algebra. https://doi.org/10.1080/03081087.2022.2117771

    Article  Google Scholar 

  • Chang SY, Wei Y (2022a) T-product tensors-part II: tail bounds for sums of random T-product tensors. Comput Appl Math 41(3):1–32

    MathSciNet  MATH  Google Scholar 

  • Chang SY, Wei Y (2022b) T-square tensors-Part I: inequalities. Comput Appl Math 41(1):1–27

    MathSciNet  Google Scholar 

  • Chang SY, Wei Y (2022c) Tail bounds for random tensors summation: majorization approach. J Comput Appl Math 416:25 (Id/No 114533)

    MathSciNet  MATH  Google Scholar 

  • Chang K-C, Pearson K, Zhang T (2009) On eigenvalue problems of real symmetric tensors. J Math Anal Appl 350(1):416–422

    MathSciNet  MATH  Google Scholar 

  • Che M, Li G, Qi L, Wei Y (2017) Pseudo-spectra theory of tensors and tensor polynomial eigenvalue problems. Linear Algebra Appl 533:536–572

    MathSciNet  MATH  Google Scholar 

  • Chen C, Surana A, Bloch A, Rajapakse I (2019) Multilinear time invariant system theory. In: 2019 Proceedings of the Conference on control and its applications, SIAM, pp 118–125

  • Chen C, Surana A, Bloch AM, Rajapakse I (2021) Multilinear control systems theory. SIAM J Control Optim 59(1):749–776

    MathSciNet  MATH  Google Scholar 

  • Conway JB (2019) A course in functional analysis. Springer, New York

    Google Scholar 

  • Cui L-B, Chen C, Li W, Ng MK (2016) An eigenvalue problem for even order tensors with its applications. Linear Multilinear Algebra 64(4):602–621

    MathSciNet  MATH  Google Scholar 

  • De Lathauwer L, Castaing J, Cardoso J-F (2007) Fourth-order cumulant-based blind identification of underdetermined mixtures. IEEE Trans Signal Process 55(6):2965–2973

    MathSciNet  MATH  Google Scholar 

  • Demmel J (1992) The componentwise distance to the nearest singular matrix. SIAM J Matrix Anal Appl 13(1):10–19

    MathSciNet  MATH  Google Scholar 

  • Ding W, Wei Y (2015) Generalized tensor eigenvalue problems. SIAM J Matrix Anal Appl 36(3):1073–1099

    MathSciNet  MATH  Google Scholar 

  • Dolgov S, Kalise D, Kunisch KK (2021) Tensor decomposition methods for high-dimensional Hamilton-Jacobi-Bellman equations. SIAM J Sci Comput 43(3):A1625–A1650

    MathSciNet  MATH  Google Scholar 

  • Du K, Wei Y (2006) Structured pseudospectra and structured sensitivity of eigenvalues. J Comput Appl Math 197(2):502–519

    MathSciNet  MATH  Google Scholar 

  • Eising R (1984) Between controllable and uncontrollable. Syst Control Lett 4(5):263–264

    MathSciNet  MATH  Google Scholar 

  • Elsner L, Sun J-G (1982) Perturbation theorems for the generalized eigenvalue problem. Linear Algebra Appl 48:341–357

    MathSciNet  MATH  Google Scholar 

  • Frayssé V, Toumazou V (1998) A note on the normwise perturbation theory for the regular generalized eigenproblem. Numer Linear Algebra Appl 5(1):1–10

    MathSciNet  MATH  Google Scholar 

  • Golub GH, Van Loan CF (2013) Matrix computations. Johns Hopkins University Press, Baltimore

    MATH  Google Scholar 

  • Gu M (2000) New methods for estimating the distance to uncontrollability. SIAM J Matrix Anal Appl 21(3):989–1003

    MathSciNet  MATH  Google Scholar 

  • He J, Li C, Wei Y (2020) Pseudospectra localization sets of tensors with applications. J Comput Appl Math 369:19

    MathSciNet  MATH  Google Scholar 

  • Higham DJ, Higham NJ (1998) Structured backward error and condition of generalized eigenvalue problems. SIAM J Matrix Anal Appl 20(2):493–512

    MathSciNet  MATH  Google Scholar 

  • Higham NJ, Tisseur F (2002) More on pseudospectra for polynomial eigenvalue problems and applications in control theory. Linear Algebra Appl 351:435–453

    MathSciNet  MATH  Google Scholar 

  • Hillar CJ, Lim L-H (2013) Most tensor problems are NP-hard. J ACM 60(6):1–39

    MathSciNet  MATH  Google Scholar 

  • Horn RA, Johnson CR (2012) Matrix analysis. Cambridge University Press, New York

    Google Scholar 

  • Itskov M (2000) On the theory of fourth-order tensors and their applications in computational mechanics. Comput Methods Appl Mech Eng 189(2):419–438

    MathSciNet  MATH  Google Scholar 

  • Ji J, Wei Y (2018) The Drazin inverse of an even-order tensor and its application to singular tensor equations. Comput Math Appl 75(9):3402–3413

    MathSciNet  MATH  Google Scholar 

  • Khoromskij BN (2015) Tensor numerical methods for multidimensional PDEs: theoretical analysis and initial applications. ESAIM Proc Surv 48:1–28

    MathSciNet  MATH  Google Scholar 

  • Kolda TG, Bader BW (2009) Tensor decompositions and applications. SIAM Rev 51(3):455–500

    MathSciNet  MATH  Google Scholar 

  • Lancaster P, Psarrakos P (2005) On the pseudospectra of matrix polynomials. SIAM J Matrix Anal Appl 27(1):115–129

    MathSciNet  MATH  Google Scholar 

  • Li W, Ng MK (2014) On the limiting probability distribution of a transition probability tensor. Linear Multilinear Algebra 62(3):362–385

    MathSciNet  MATH  Google Scholar 

  • Li C, Liu Q, Wei Y (2019) Pseudospectra localizations for generalized tensor eigenvalues to seek more positive definite tensors. Comput Appl Math 38(4):22

    MathSciNet  MATH  Google Scholar 

  • Liang M, Zheng B (2019) Further results on Moore-Penrose inverses of tensors with application to tensor nearness problems. Comput Math Appl 77(5):1282–1293

    MathSciNet  MATH  Google Scholar 

  • Liang M, Zheng B, Zhao R (2019) Tensor inversion and its application to the tensor equations with Einstein product. Linear Multilinear Algebra 67(4):843–870

    MathSciNet  MATH  Google Scholar 

  • Lim L-H (2005) Singular values and eigenvalues of tensors: a variational approach. In: 1st IEEE International Workshop on computational advances in multi-sensor adaptive processing, pp 129–132

  • Liu W, Jin X (2021) A study on T-eigenvalues of third-order tensors. Linear Algebra Appl 612:357–374

    MathSciNet  MATH  Google Scholar 

  • Ma H, Li N, Stanimirović PS, Katsikis VN (2019) Perturbation theory for Moore-Penrose inverse of tensor via Einstein product. Comput Appl Math 38(3):1–24

    MathSciNet  MATH  Google Scholar 

  • Malyshev AN, Sadkane M (2004) Componentwise pseudospectrum of a matrix. Linear Algebra Appl 378:283–288

    MathSciNet  MATH  Google Scholar 

  • Mehrabadi MM, Cowin SC (1990) Eigentensors of linear anisotropic elastic materials. Q J Mech Appl Math 43(1):15–41

    MathSciNet  MATH  Google Scholar 

  • Miao Y, Qi L, Wei Y (2020) Generalized tensor function via the tensor singular value decomposition based on the T-product. Linear Algebra Appl 590:258–303

    MathSciNet  MATH  Google Scholar 

  • Miao Y, Qi L, Wei Y (2021) T-Jordan canonical form and T-Drazin inverse based on the T-product. Commun Appl Math Comput 3(2):201–220

    MathSciNet  MATH  Google Scholar 

  • Miao Y, Wei Y, Chen Z (2022) Fourth-order tensor Riccati equations with the Einstein product. Linear Multilinear Algebra 70(10):1831–1853

    MathSciNet  MATH  Google Scholar 

  • Moler CB, Stewart GW (1973) An algorithm for generalized matrix eigenvalue problems. SIAM J Numer Anal 10(2):241–256

    MathSciNet  MATH  Google Scholar 

  • Poljak S, Rohn J (1993) Checking robust nonsingularity is NP-hard. Math Control Signals Syst 6(1):1–9

    MathSciNet  MATH  Google Scholar 

  • Qi L (2005) Eigenvalues of a real supersymmetric tensor. J Symb Comput 40(6):1302–1324

    MathSciNet  MATH  Google Scholar 

  • Ragnarsson S, Van Loan CF (2012) Block tensor unfoldings. SIAM J Matrix Anal Appl 33(1):149–169

    MathSciNet  MATH  Google Scholar 

  • Rogers M, Li L, Russell SJ (2013) Multilinear dynamical systems for tensor time series. Adv Neural Inf Process Syst 26:2634–2642

    Google Scholar 

  • Rump SM (1999) Ill-conditioned matrices are componentwise near to singularity. SIAM Rev 41(1):102–112

    MathSciNet  MATH  Google Scholar 

  • Rump SM (2003) Perron-Frobenius theory for complex matrices. Linear Algebra Appl 363:251–273

    MathSciNet  MATH  Google Scholar 

  • Shi X, Wei Y (2012) A sharp version of Bauer-Fike’s theorem. J Comput Appl Math 236(13):3218–3227

    MathSciNet  MATH  Google Scholar 

  • Song Y, Qi L (2013) Spectral properties of positively homogeneous operators induced by higher order tensors. SIAM J Matrix Anal Appl 34(4):1581–1595

    MathSciNet  MATH  Google Scholar 

  • Stewart GW, Sun J (1990) Matrix perturbation theory. Academic Press, Boston

    MATH  Google Scholar 

  • Sun L, Zheng B, Bu C, Wei Y (2016) Moore-Penrose inverse of tensors via Einstein product. Linear Multilinear Algebra 64(4):686–698

    MathSciNet  MATH  Google Scholar 

  • Thomas GW (2002) Eigtool. Version 2.1 (beta), 16 March 2009. http://www.comlab.ox.ac.uk/pseudospectra/eigtool/

  • Trefethen LN, Embree M (2005) Spectra and pseudospectra: the behavior of nonnormal matrices and operators. Princeton University Press, Princeton

    MATH  Google Scholar 

  • Varga RS (2004) Geršgorin and his circles, vol 36. Springer, Berlin

    MATH  Google Scholar 

  • Vasilescu M, Terzopoulos D (2003) Multilinear subspace analysis of image ensembles. In: 2003 IEEE Computer Society Conference on computer vision and pattern recognition, volume 2, pp 93–99

  • Wei Y, Stanimirović P, Petković M (2018) Numerical and symbolic computations of generalized inverses. World Scientific, Hackensack

    MATH  Google Scholar 

  • Zhang G, Li H, Wei Y (2022) Componentwise perturbation analysis for the generalized Schur decomposition. Calcolo 59(2):1–32

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the handling editor Jinyun Yuan and two referees for their detailed comments.

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China

    Yuchao Wang

  2. School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, People’s Republic of China

    Yimin Wei

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  1. Yuchao Wang
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Correspondence to Yimin Wei.

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Communicated by Jinyun Yuan.

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Y. Wang is supported by the National Natural Science Foundation of China under Grant 12271108 and Shanghai Municipal Science and Technology Commission under Grant 22WZ2501900.

Y. Wei is supported by the National Natural Science Foundation of China under Grant 12271108 and the Innovation Program of Shanghai Municipal Education Committee.

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Wang, Y., Wei, Y. Generalized eigenvalue for even order tensors via Einstein product and its applications in multilinear control systems. Comp. Appl. Math. 41, 419 (2022). https://doi.org/10.1007/s40314-022-02129-1

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