April  2019, 15(2): 775-789. doi: 10.3934/jimo.2018070

Partially symmetric nonnegative rectangular tensors and copositive rectangular tensors

School of Mathematical Sciences, Tianjin University, Tianjin 300350, China

* Corresponding author: Wei Wu

Received  September 2016 Revised  December 2017 Published  April 2019 Early access  June 2018

Fund Project: This work was supported by NSF grant of China (Grant No. 11371276).

In this paper, we prove a maximum property of the largest H-singular value of a partially symmetric nonnegative rectangular tensor, and establish some bounds for this singular value. Then we give the definition of copositive rectangular tensors. This concept extends from the concept of copositive square tensors. Partially symmetric nonnegative rectangular tensors and positive semi-definite rectangular tensors are examples of copositive rectangular tensors. We establish some necessary conditions and some sufficient conditions for a real partially symmetric rectangular tensor to be a copositive rectangular tensor. We also give an equivalent definition of strictly copositive rectangular tensors. Moreover, some further properties of copositive rectangular tensors are discussed.

Citation: Yining Gu, Wei Wu. Partially symmetric nonnegative rectangular tensors and copositive rectangular tensors. Journal of Industrial and Management Optimization, 2019, 15 (2) : 775-789. doi: 10.3934/jimo.2018070
References:
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L. Bloy and R. Verma, On computing the underlying fiber directions from the diffusion orientation distribution function, In: Medical Image Computing and Computer-Assisted Intervention-MICCAI 2008, Springer, Berlin/Heidelberg, (2008), 1-8.  doi: 10.1007/978-3-540-85988-8_1.

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K. C. ChangL. Qi and G. Zhou, Singular values of real rectangular tensor, J. Math Anal. Appl., 370 (2010), 284-294.  doi: 10.1016/j.jmaa.2010.04.037.

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K. C. ChangK. Pearson and T. Zhang, Perron Frobenius Theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), 507-520.  doi: 10.4310/CMS.2008.v6.n2.a12.

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D. DahlJ. M. LeinassJ. Myrheim and E. Ovrum, A tensor product matrix approximation problem in quantum physics, Linear Algebra Appl., 420 (2007), 711-725.  doi: 10.1016/j.laa.2006.08.026.

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L. De LathauwerB. D. Moor and J. Vandewalle, On the best rank-1 and rank-$ (R_{1},R_{2},...,R_{N})$ approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), 1324-1342.  doi: 10.1137/S0895479898346995.

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L. H. Lim, Singular values and eigenvalues of tensors: A variational approach, in CAMSAP'05: Proceeding of the IEEE International Workshop on Computational Advances in Multi-tensor Adaptive Processing, 2005, 129–132.

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C. LingJ. NieL. Qi and Y. Ye, SDP and SOS relaxations for bi-quadratic optimization over unit spheres, SIAM J. Optim., 20 (2009), 1286-1310.  doi: 10.1137/080729104.

[11]

M. NgL. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SLAM J. Matrix Anal. Appl., 31 (2009), 1090-1099.  doi: 10.1137/09074838X.

[12]

Q. NiL. Qi and F. Wang, An eigenvalue method for the positive definition identification problem, IEEE Transactions on Automatic Control, 53 (2008), 1096-1107.  doi: 10.1109/TAC.2008.923679.

[13]

L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.

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L. QiW. Sun and Y. Wang, Numerical multilinear algebra and its applications, Front. Math. China, 2 (2007), 501-526.  doi: 10.1007/s11464-007-0031-4.

[15]

L. QiY. Wang and E. X. Wu, D-eigenvalues of diffusion kurtosis tensor, Journal of Computational and Applied Mathematics, 221 (2008), 150-157.  doi: 10.1016/j.cam.2007.10.012.

[16]

L. Qi, Symmetric nonnegative tensors and copositive tensors, Linear Algebra Appl., 439 (2013), 228-238.  doi: 10.1016/j.laa.2013.03.015.

[17]

L. QiH. H. Dai and D. Han, Conditions for strong ellipticity and $M$-eigenvalues, Front. Math. China, 4 (2009), 349-364.  doi: 10.1007/s11464-009-0016-6.

[18]

P. Rosakis, Ellipticity and deformations with discontinuous deformation gradients in finite elastostatics, Arch. Ration. Mech. Anal., 109 (1990), 1-37.  doi: 10.1007/BF00377977.

[19]

E. Schrödinger, Die gegenwärtige situation in der quantenmechanik, Naturwissenschaften, Naturwissenschaften, 23 (1935), 807-812,823-828,844-849.

[20]

H. C. Simpson and S. J. Spector, On copositive matrices and strong ellipticity for isotropic elastic materials, Arch. Ration. Mech. Anal., 84 (1983), 55-68.  doi: 10.1007/BF00251549.

[21]

Y. Song and L. Qi, Necessary and sufficient conditions for copositive tensors, Linear Multilinear Algebra, 63 (2015), 120-131.  doi: 10.1080/03081087.2013.851198.

[22]

Y. Wang and M. Aron, A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media, J. Elasticity, 44 (1996), 89-96.  doi: 10.1007/BF00042193.

[23]

Y. WangL. Qi and X. Zhang, A practical method for computing the largest $M$-eigenvalue of a fourth-order partially symmetric tensor, Numerical Linear Algebra with Applications, 16 (2009), 589-601.  doi: 10.1002/nla.633.

[24]

Y. N. Yang and Q. Yang, Singular values of nonnegative rectangular tensors, Front. Math. China, 6 (2011), 363-378.  doi: 10.1007/s11464-011-0108-y.

[25]

L. ZhangL. QiZ. Luo and Y. Xu, The dominant eigenvalue of an essentially nonnegative tensor, Numerical Linear Algebra with Applications, 20 (2013), 929-941.  doi: 10.1002/nla.1880.

show all references

References:
[1]

L. Bloy and R. Verma, On computing the underlying fiber directions from the diffusion orientation distribution function, In: Medical Image Computing and Computer-Assisted Intervention-MICCAI 2008, Springer, Berlin/Heidelberg, (2008), 1-8.  doi: 10.1007/978-3-540-85988-8_1.

[2]

K. C. ChangL. Qi and G. Zhou, Singular values of real rectangular tensor, J. Math Anal. Appl., 370 (2010), 284-294.  doi: 10.1016/j.jmaa.2010.04.037.

[3]

K. C. ChangK. Pearson and T. Zhang, Perron Frobenius Theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), 507-520.  doi: 10.4310/CMS.2008.v6.n2.a12.

[4]

D. DahlJ. M. LeinassJ. Myrheim and E. Ovrum, A tensor product matrix approximation problem in quantum physics, Linear Algebra Appl., 420 (2007), 711-725.  doi: 10.1016/j.laa.2006.08.026.

[5]

L. De LathauwerB. D. Moor and J. Vandewalle, On the best rank-1 and rank-$ (R_{1},R_{2},...,R_{N})$ approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), 1324-1342.  doi: 10.1137/S0895479898346995.

[6]

A. EinsteinB. Podolsky and N. Rosen, Can quantum-mechanical description of physical reality be considered complete, Phys. Rev., 47 (1995), 777-780.  doi: 10.1103/PhysRev.47.777.

[7]

J. K. Knowles and E. Sternberg, On the ellipticity of the equations of non-linear elastostatics for a special material, J. Elasticity, 5 (1975), 341-361.  doi: 10.1007/BF00126996.

[8]

J. K. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain, Arch. Ration. Mech. Anal., 63 (1997), 321-336.  doi: 10.1007/BF00279991.

[9]

L. H. Lim, Singular values and eigenvalues of tensors: A variational approach, in CAMSAP'05: Proceeding of the IEEE International Workshop on Computational Advances in Multi-tensor Adaptive Processing, 2005, 129–132.

[10]

C. LingJ. NieL. Qi and Y. Ye, SDP and SOS relaxations for bi-quadratic optimization over unit spheres, SIAM J. Optim., 20 (2009), 1286-1310.  doi: 10.1137/080729104.

[11]

M. NgL. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SLAM J. Matrix Anal. Appl., 31 (2009), 1090-1099.  doi: 10.1137/09074838X.

[12]

Q. NiL. Qi and F. Wang, An eigenvalue method for the positive definition identification problem, IEEE Transactions on Automatic Control, 53 (2008), 1096-1107.  doi: 10.1109/TAC.2008.923679.

[13]

L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.

[14]

L. QiW. Sun and Y. Wang, Numerical multilinear algebra and its applications, Front. Math. China, 2 (2007), 501-526.  doi: 10.1007/s11464-007-0031-4.

[15]

L. QiY. Wang and E. X. Wu, D-eigenvalues of diffusion kurtosis tensor, Journal of Computational and Applied Mathematics, 221 (2008), 150-157.  doi: 10.1016/j.cam.2007.10.012.

[16]

L. Qi, Symmetric nonnegative tensors and copositive tensors, Linear Algebra Appl., 439 (2013), 228-238.  doi: 10.1016/j.laa.2013.03.015.

[17]

L. QiH. H. Dai and D. Han, Conditions for strong ellipticity and $M$-eigenvalues, Front. Math. China, 4 (2009), 349-364.  doi: 10.1007/s11464-009-0016-6.

[18]

P. Rosakis, Ellipticity and deformations with discontinuous deformation gradients in finite elastostatics, Arch. Ration. Mech. Anal., 109 (1990), 1-37.  doi: 10.1007/BF00377977.

[19]

E. Schrödinger, Die gegenwärtige situation in der quantenmechanik, Naturwissenschaften, Naturwissenschaften, 23 (1935), 807-812,823-828,844-849.

[20]

H. C. Simpson and S. J. Spector, On copositive matrices and strong ellipticity for isotropic elastic materials, Arch. Ration. Mech. Anal., 84 (1983), 55-68.  doi: 10.1007/BF00251549.

[21]

Y. Song and L. Qi, Necessary and sufficient conditions for copositive tensors, Linear Multilinear Algebra, 63 (2015), 120-131.  doi: 10.1080/03081087.2013.851198.

[22]

Y. Wang and M. Aron, A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media, J. Elasticity, 44 (1996), 89-96.  doi: 10.1007/BF00042193.

[23]

Y. WangL. Qi and X. Zhang, A practical method for computing the largest $M$-eigenvalue of a fourth-order partially symmetric tensor, Numerical Linear Algebra with Applications, 16 (2009), 589-601.  doi: 10.1002/nla.633.

[24]

Y. N. Yang and Q. Yang, Singular values of nonnegative rectangular tensors, Front. Math. China, 6 (2011), 363-378.  doi: 10.1007/s11464-011-0108-y.

[25]

L. ZhangL. QiZ. Luo and Y. Xu, The dominant eigenvalue of an essentially nonnegative tensor, Numerical Linear Algebra with Applications, 20 (2013), 929-941.  doi: 10.1002/nla.1880.

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