The Sign-Real Spectral Radius for Real Tensors

Document Type : Research Paper


1 Vali-e-Asr University of Rafsanjan

2 Department of Mathematics, Faculty of Sciences, Imam Hossein Comprehensive University, Tehran, Islamic Republic of Iran



In this paper a new quantity for real tensors, the sign-real spectral radius, is defined and investigated. Various characterizations, bounds and some properties are derived. In certain aspects our quantity shows similar behavior to the spectral radius of a nonnegative tensor. In fact, we generalize the Perron Frobenius theorem for nonnegative tensors to the class of real tensors.


[1] J.F. Cardoso, High-order contrasts for independent component analysis, Neural Comput., 11 (1999), 157-192.
[2] F. Chaitin-Chatelin and V. Frays, Lectures on finite precision computations, Siam Ser. Software Environments Tools, Philadelphia, 1996.
[3] K.C. Chang, K. Pearson and T. Zhang, On eigenvalue problems of real symmetric tensors, J. Math. Anal. Appl., 350(1) (2009), 416-422.
[4] K.C. Chang, K. Pearson and T. Zhang, Perron Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6(2) (2008), 507-520.
[5] P. Comon, Independent component analysis, a new concept?, Signal Process., 36(3) (1994), 287-314.
[6] P. Comon and B. Mourrain, Decomposition of quantics in sums of powers of linear forms, Signal Process., 53(2-3) (1996), 96-107.
[7] D. Cox, J. Little and D. O'Shea, Using algebraic geometry, Springer-Verlag, NewYork, 1998.
[8] L. De Lathauwer and B. De Moor, From matrix to tensor: Multilinear algebra and signal processing,
Institute of mathematics and its applications conference series, 67 (1998), 1-16.
[9] L. De Lathauwer, B. De Moor and J. Vandewalle, A multilinear singular value decomposition, SIAM J. Matrix Anal. Appl., 21(4) (2001), 1253-1278.
[10] W. Ding and Y. Wei, Generalized tensor eigenvalue problems, SIAM J. Matrix Anal. Appl., 36(3) (2015), 1073-1099.
[11] P. Drineas and L.H. Lim, A multilinear spectral theory of hypergraphs and expander hypergraphs, 2005.
[12] S. Friedland, S. Gaubert and L. Han, Perron Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra Appl., 438(2) (2013), 738-749.
[13] N.J. Higham, Accuracy and stability of numerical algorithms, Siam, Philadelphia, 1996.
[14] S. Hu, Z. Huang, C. Ling and L. Qi, On determinants and eigenvalue theory of tensors, J. Symb. Comput., 50 (2013), 508-531.
[15] K.C. Huang, M.D. Xue and M.W. Lu, Tensor Analysis, Second ed., Tsinghua Univ. Publisher, Beijing, Chinese, 2003.
[16] L.H. Lim, Multilinear pagerank: measuring higher order connectivity in linked objects, The Internet: Today and Tomorrow, 2005.
[17] L.H. Lim, Singular values and eigenvalues of tensors, a variational approach, Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, (2005), 129-132.
[18] P. McCullagh, Tensor methods in statistics, Monographs on Statistics and Applied Probability, Chapman and Hall, 1987.
[19] Q. Ni, L. Qi and F. Wang, An eigenvalue method for the positive definiteness identification problem, Department of Applied Mathematics, The Hong Kong Polytechnic University, 2005.
[20] M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of anonnegative tensor, SIAM J. Matrix Anal. Appl., 31(3)  (2009), 1090-1099.
[21] L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40(6) (2005), 1302-1324.
[22] J. Rohn, Systems of linear interval equations, Linear Algerba Appl., 126 (1989), 39-78.
[23] S.M. Rump, Theorems of Perron Frobenius type for matrices without sign restrictions, Linear Algebra Appl.}, 266 (1997), 1-42.
[24] J.Y. Shao, A general product of tensors with applications, Linear Algebra Appl., 439(8) (2013), 2350-2366.
[25] Y.R. Talpaert, Tensor analysis and continuum mechanics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002.
[26] Y. Yang and Q. Yang, Further results for Perron Frobenius theorem for nonnegative tensors, SIAM J. Matrix Anal. Appl., 31(5) (2010), 2517-2530.
[27] T. Zhang and G.H. Golub, Rank-One Approximation to High Order Tensors, SIAM J. Matrix Anal. Appl., 23(2) (2001), 534-550.