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October  2015, 11(4): 1263-1274. doi: 10.3934/jimo.2015.11.1263

Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors

Haibin Chen 1, and  Liqun Qi 1,

1. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China, China

Received  May 2014 Revised  September 2014 Published  March 2015

Motivated by symmetric Cauchy matrices, we define symmetric Cauchy tensors and their generating vectors in this paper. Hilbert tensors are symmetric Cauchy tensors. An even order symmetric Cauchy tensor is positive semi-definite if and only if its generating vector is positive. An even order symmetric Cauchy tensor is positive definite if and only if its generating vector has positive and mutually distinct entries. This extends Fiedler's result for symmetric Cauchy matrices to symmetric Cauchy tensors. Then, it is proven that the positive semi-definiteness character of an even order symmetric Cauchy tensor can be equivalently checked by the monotone increasing property of a homogeneous polynomial related to the Cauchy tensor. The homogeneous polynomial is strictly monotone increasing in the nonnegative orthant of the Euclidean space when the even order symmetric Cauchy tensor is positive definite. At last, bounds of the largest H-eigenvalue of a positive semi-definite symmetric Cauchy tensor are given and several spectral properties on Z-eigenvalues of odd order symmetric Cauchy tensors are shown. Further questions on Cauchy tensors are raised.
Keywords: generating vector., Cauchy tensor, eigenvalue, positive definiteness, Positive semi-definiteness.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Haibin Chen, Liqun Qi. Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1263-1274. doi: 10.3934/jimo.2015.11.1263
References:
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K. C. Chang, K. Pearson and T. Zhang, Perron Frobenius theorem for nonnegative tensors, Commu. Math. Sci., 6 (2008), 507-520. doi: 10.4310/CMS.2008.v6.n2.a12.  Google Scholar

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W. Ding, L. Qi and Y. Wei, M-Tensors and nonsingular M-tensors, Lin. Alg. Appl., 439 (2013), 3264-3278. doi: 10.1016/j.laa.2013.08.038.  Google Scholar

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M. Fiedler, Notes on Hilbert and Cauchy matrices, Lin. Alg. Appl., 432 (2010), 351-356. doi: 10.1016/j.laa.2009.08.014.  Google Scholar

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T. Finck, G. Heinig and K. Rost, An inversion formula and fast algorithms for Cauchy-Vandermonde matrices, Lin. Alg. Appl., 183 (1993), 179-191. doi: 10.1016/0024-3795(93)90431-M.  Google Scholar

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J. He and T. Z. Huang, Inequalities for M-tensors, Journal of Inequality and Applications, 2014 (2014), p114. doi: 10.1186/1029-242X-2014-114.  Google Scholar

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G. Heinig, Inversion of generalized Cauchy matrices and other classes of structured matrices, Linear Algebra for Signal Processing, Springer, New York, (1995), 63-81. doi: 10.1007/978-1-4612-4228-4_5.  Google Scholar

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G. Pólya and G. Szegö, Zweiter Band, Springer, Berlin, 1925. Google Scholar

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

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L. Qi, $H^+$-eigenvalues of Laplacian and signless Laplacian tensors, Communications in Mathematical Sciences, 12 (2014), 1045-1064. doi: 10.4310/CMS.2014.v12.n6.a3.  Google Scholar

[13]

L. Qi, Hankel tensors: Associated Hankel matrices and Vandermonde decomposition, Communications in Mathematical Sciences, 13 (2015), 113-125. doi: 10.4310/CMS.2015.v13.n1.a6.  Google Scholar

[14]

L. Qi and Y. Song, An even order symmetric B tensor is positive definite, Lin. Alg. Appl., 457 (2014), 303-312. doi: 10.1016/j.laa.2014.05.026.  Google Scholar

[15]

L. Qi, C. Xu and Y. Xu, Nonnegative tensor factorization, completely positive tensors and an hierarchical elimination algorithm, SIAM J. Matrix Anal. Appl., 35 (2014), 1227-1241. doi: 10.1137/13092232X.  Google Scholar

[16]

S. Solak and D. Bozkruk, On the spectral norms of Cauchy-Toeplitz and Cauchy-Hankel matrices, Appl. Math. Comput., 140 (2003), 231-238. doi: 10.1016/S0096-3003(02)00205-9.  Google Scholar

[17]

Y. Song and L. Qi, Some properties of infinite and finite dimension Hilbert tensors, Lin. Alg. Appl., 451 (2014), 1-14. Google Scholar

[18]

Y. Song and L. Qi, Properties of some classes of structured tensors, J. Optim. Theory Appl., (2015), DOI 10.1007/s10957-014-0616-5. doi: 10.1007/s10957-014-0616-5.  Google Scholar

[19]

E. E. Tyrtyshnikov, Cauchy-Toeplitz matrices and some applications, Lin. Alg. Appl., 149 (1991), 1-18. doi: 10.1016/0024-3795(91)90321-M.  Google Scholar

[20]

E. E. Tyrtyshnikov, Singular values of Cauchy-Toeplitz matrices, Lin. Alg. Appl., 161 (1992), 99-116. doi: 10.1016/0024-3795(92)90007-W.  Google Scholar

[21]

P. Yuan and L. You, Some remarks on P, P$_0$, B and B$_0$ tensors, Lin. Alg. Appl., 459 (2014) 511-521. Google Scholar

[22]

L. Zhang, L. Qi and G. Zhou, M-tensors and some applications, SIAM J. Matrix Anal. Appl., 35 (2014), 437-452. doi: 10.1137/130915339.  Google Scholar

show all references

References:
[1]

K. C. Chang, K. Pearson and T. Zhang, Perron Frobenius theorem for nonnegative tensors, Commu. Math. Sci., 6 (2008), 507-520. doi: 10.4310/CMS.2008.v6.n2.a12.  Google Scholar

[2]

Z. Chen and L. Qi, Circulant tensors with applications to spectral hypergraph theory and stochastic process, preprint, 2014, arXiv:1312.2752. Google Scholar

[3]

W. Ding, L. Qi and Y. Wei, M-Tensors and nonsingular M-tensors, Lin. Alg. Appl., 439 (2013), 3264-3278. doi: 10.1016/j.laa.2013.08.038.  Google Scholar

[4]

W. Ding, L. Qi and Y. Wei, Fast Hankel tensor-vector products and application to exponential data fitting, Numer. Lin. Alg. Appl., (2015), DOI: 10.1002/nla.1970. doi: 10.1002/nla.1970.  Google Scholar

[5]

M. Fiedler, Notes on Hilbert and Cauchy matrices, Lin. Alg. Appl., 432 (2010), 351-356. doi: 10.1016/j.laa.2009.08.014.  Google Scholar

[6]

T. Finck, G. Heinig and K. Rost, An inversion formula and fast algorithms for Cauchy-Vandermonde matrices, Lin. Alg. Appl., 183 (1993), 179-191. doi: 10.1016/0024-3795(93)90431-M.  Google Scholar

[7]

I. Gohberg and V. Olshevsky, Fast algorithms with preprocessing for matrix-vector multiplication problems, J. Complexity, 10 (1994), 411-427. doi: 10.1006/jcom.1994.1021.  Google Scholar

[8]

J. He and T. Z. Huang, Inequalities for M-tensors, Journal of Inequality and Applications, 2014 (2014), p114. doi: 10.1186/1029-242X-2014-114.  Google Scholar

[9]

G. Heinig, Inversion of generalized Cauchy matrices and other classes of structured matrices, Linear Algebra for Signal Processing, Springer, New York, (1995), 63-81. doi: 10.1007/978-1-4612-4228-4_5.  Google Scholar

[10]

G. Pólya and G. Szegö, Zweiter Band, Springer, Berlin, 1925. Google Scholar

[11]

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

[12]

L. Qi, $H^+$-eigenvalues of Laplacian and signless Laplacian tensors, Communications in Mathematical Sciences, 12 (2014), 1045-1064. doi: 10.4310/CMS.2014.v12.n6.a3.  Google Scholar

[13]

L. Qi, Hankel tensors: Associated Hankel matrices and Vandermonde decomposition, Communications in Mathematical Sciences, 13 (2015), 113-125. doi: 10.4310/CMS.2015.v13.n1.a6.  Google Scholar

[14]

L. Qi and Y. Song, An even order symmetric B tensor is positive definite, Lin. Alg. Appl., 457 (2014), 303-312. doi: 10.1016/j.laa.2014.05.026.  Google Scholar

[15]

L. Qi, C. Xu and Y. Xu, Nonnegative tensor factorization, completely positive tensors and an hierarchical elimination algorithm, SIAM J. Matrix Anal. Appl., 35 (2014), 1227-1241. doi: 10.1137/13092232X.  Google Scholar

[16]

S. Solak and D. Bozkruk, On the spectral norms of Cauchy-Toeplitz and Cauchy-Hankel matrices, Appl. Math. Comput., 140 (2003), 231-238. doi: 10.1016/S0096-3003(02)00205-9.  Google Scholar

[17]

Y. Song and L. Qi, Some properties of infinite and finite dimension Hilbert tensors, Lin. Alg. Appl., 451 (2014), 1-14. Google Scholar

[18]

Y. Song and L. Qi, Properties of some classes of structured tensors, J. Optim. Theory Appl., (2015), DOI 10.1007/s10957-014-0616-5. doi: 10.1007/s10957-014-0616-5.  Google Scholar

[19]

E. E. Tyrtyshnikov, Cauchy-Toeplitz matrices and some applications, Lin. Alg. Appl., 149 (1991), 1-18. doi: 10.1016/0024-3795(91)90321-M.  Google Scholar

[20]

E. E. Tyrtyshnikov, Singular values of Cauchy-Toeplitz matrices, Lin. Alg. Appl., 161 (1992), 99-116. doi: 10.1016/0024-3795(92)90007-W.  Google Scholar

[21]

P. Yuan and L. You, Some remarks on P, P$_0$, B and B$_0$ tensors, Lin. Alg. Appl., 459 (2014) 511-521. Google Scholar

[22]

L. Zhang, L. Qi and G. Zhou, M-tensors and some applications, SIAM J. Matrix Anal. Appl., 35 (2014), 437-452. doi: 10.1137/130915339.  Google Scholar

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