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Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors
1. | Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China, China |
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. |
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. |
[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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