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

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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

Abstract
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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 and Management Optimization, 2015, 11 (4) : 1263-1274. doi: 10.3934/jimo.2015.11.1263

References:

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[2]	Z. Chen and L. Qi, Circulant tensors with applications to spectral hypergraph theory and stochastic process, preprint, 2014, arXiv:1312.2752.
[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.
[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.
[5]	M. Fiedler, Notes on Hilbert and Cauchy matrices, Lin. Alg. Appl., 432 (2010), 351-356. doi: 10.1016/j.laa.2009.08.014.
[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.
[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.
[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.
[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.
[10]	G. Pólya and G. Szegö, Zweiter Band, Springer, Berlin, 1925.
[11]	L. Qi, Eigenvalue of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302-1324. doi: 10.1016/j.jsc.2005.05.007.
[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.
[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.
[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.
[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.
[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.
[17]	Y. Song and L. Qi, Some properties of infinite and finite dimension Hilbert tensors, Lin. Alg. Appl., 451 (2014), 1-14.
[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.
[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.
[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.
[21]	P. Yuan and L. You, Some remarks on P, P$_0$, B and B$_0$ tensors, Lin. Alg. Appl., 459 (2014) 511-521.
[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.

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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.
[2]	Z. Chen and L. Qi, Circulant tensors with applications to spectral hypergraph theory and stochastic process, preprint, 2014, arXiv:1312.2752.
[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.
[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.
[5]	M. Fiedler, Notes on Hilbert and Cauchy matrices, Lin. Alg. Appl., 432 (2010), 351-356. doi: 10.1016/j.laa.2009.08.014.
[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.
[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.
[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.
[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.
[10]	G. Pólya and G. Szegö, Zweiter Band, Springer, Berlin, 1925.
[11]	L. Qi, Eigenvalue of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302-1324. doi: 10.1016/j.jsc.2005.05.007.
[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.
[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.
[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.
[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.
[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.
[17]	Y. Song and L. Qi, Some properties of infinite and finite dimension Hilbert tensors, Lin. Alg. Appl., 451 (2014), 1-14.
[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.
[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.
[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.
[21]	P. Yuan and L. You, Some remarks on P, P$_0$, B and B$_0$ tensors, Lin. Alg. Appl., 459 (2014) 511-521.
[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.

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