l ^k,s-Singular values and spectral radius of rectangular tensors

Abstract

The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we study the singular values/vectors problem of real nonnegative partially symmetric rectangular tensors. We first introduce the concepts of l ^k,s-singular values/vectors of real partially symmetric rectangular tensors. Then, based upon the presented properties of l ^k,s-singular values /vectors, some properties of the related l ^k,s-spectral radius are discussed. Furthermore, we prove two analogs of Perron-Frobenius theorem and weak Perron-Frobenius theorem for real nonnegative partially symmetric rectangular tensors.

Keywords

Nonnegative rectangular tensor / l ^k,s-singular value / l ^k,s-spectral radius / irreducibility / weak irreducibility

References

[1] Chang K. C., Pearson K., Zhang T.. Perron Frobenius theorem for nonnegative tensors. Commun Math Sci, 2008, 6: 507-520

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

[3] Dahl D., Leinass J. M., Myrheim J., Ovrum E.. A tensor product matrix approximation problem in quantum physics. Linear Algebra Appl, 2007, 420: 711-725

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

[5] Friedland S., Gaubert S., Han L.. Perron-Frobenius theorem for nonnegative multilinear forms and extensions. Linear Algebra Appl, 2013, 438: 738-749

[6] Gaubert S., Gunawardena J.. The Perron-Frobenius theorem for homogeneous, monotone functions. Trans Amer Math Soc, 2004, 356: 4931-4950

[7] Knowles J. K., Sternberg E.. On the ellipticity of the equations of non-linear elastostatics for a special material. J Elasticity, 1975, 5: 341-361

[8] Knowles J. K., Sternberg E.. On the failure of ellipticity of the equations for finite elastostatic plane strain. Arch Ration Mech Anal, 1977, 63: 321-336

[9] Krein M. G., Rutman M. A.. Linear operators leaving invariant a cone in a Banach space. Uspekhi Mat Nauk, 1948, 3(1): 23

[10] Lim L -H. Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of the IEEE InternationalWorkshop on Computational Advances in Multi-Sensor Adaptive Processing, 1. 2005, 129–132

[11] Nussbaum R. D.. Convexity and log convexity for the spectral radius. Linear Algebra Appl, 1986, 73: 59-122

[12] Nussbaum R. D.. Hilbert’s Projective Metric and Iterated Nonlinear Maps. Mem Amer Math Soc, Vol 75, No 391, 1988, Providence: Amer Math Soc

[13] Qi L.. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput, 2005, 40: 1302-1324

[14] Rosakis P.. Ellipticity and deformations with discontinuous deformation gradients in finite elastostatics. Arch Ration Mech Anal, 1990, 109: 1-37

[15] Schröinger E.. Die gegenwätige situation in der quantenmechanik. Naturwissenschaften, 1935, 23: 807-812

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

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