Some note on a positive tensor and its Hadamard powers eigenvalue

Positivity - 2022
Mehri Pakmanesh1, Hamidreza Afshin1
1Vali-asr of Rafsanjan, Rafsanjan, Iran

Tóm tắt

In this paper, we prove that Hadamard power of a even-order positive tensor has only one positive eigenvalue. We define a conditionally negative definite tensor and an m-invertible tensor. We prove that if $${\mathscr {A}} =(a_{i_1i_2\ldots i_m})$$ is a conditionally negative definite and positive tensor, then $${\mathscr {A}}$$ has exactly one positive eigenvalue. Next, we prove that if $${\mathscr {A}} =(a_{i_1i_2\ldots i_m})$$ is an orthogonally decomposable tensor whose all entries are positive and has only one positive eigenvalue, then $${\mathscr {A}}^{\circ r} = (a^r_{i_1i_2\ldots i_m})$$ also has only one positive eigenvalue for all $$r\in [0,1)$$ . Also, the Hadamard inverse of $${\mathscr {A}}$$ (with $$a_{i_1\dots i_m}\ne 0$$ for all $$1\le i_1,\ldots , i_m\le n$$ ) denoted by $${{\mathscr {A}}^\circ }^{(-1)}=(1/a_{i_1\dots i_m})$$ is positive semi-definite.

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Tài liệu tham khảo

Ding, W., Wei, Y.: Generalized tensor eigenvalue problems. SIAM J. Matrix Anal. Appl. 36, 1073–1099 (2015)

Kolda, T., Bader, B.: Tensor decompositions and applications. SIAM Rev. 51, 455–500 (2009)

Kolda, T., Mayo, J.: Shifted power method for computing tensor eigenpairs. SIAM J. Matrix Anal. Appl. 31, 1095–1124 (2011)

Ng, M., Qi, L., Zhou, G.: Finding the largest eigenvalue of a nonnegative tensor. SIAM J. Matrix Anal. Appl. 31, 1096–1107 (2009)

Qi, L.: Symmetric nonnegative tensors and copositive tensors. Linear Algebra Appl. 439, 228–238 (2013)

Crouzeix, J.P., Ferland, J.: Criteria for quasiconvexity and pseudoconvexity: relationships and comparisons. Math. Program. 23(2), 193–205 (1982)

Donoghue, W.F.: Monotone Matrix Functions and Analytic Continuation. Springer, New York (1974)

Ferland, J.A.: Matrix-theoretic criteria for the quasiconvexity of twice continuously differentiable functions. Linear Algebra Appl. 38, 51–63 (1981)

Fiedler, M., Markham, T.L.: An observation on the Hadamard product of Hermitian matrices. Linear Algebra Appl. 215, 179–182 (1995)

Gower, J.C.: Euclidean distance geometry. Math. Sci. 7, 1–14 (1982)

Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)

Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)

Kelly, J.B.: Hypermetric spaces and metric transforms. In: Shisha, O. (ed.) Inequalities III, pp. 149–158. Academic Press, New York (1972)

Zhou, L., Liu, J., Zhu, L.: The closure property of H-tensors under the Hadamard product. J. Inequal. Appl. 2017, 231 (2017). https://doi.org/10.1186/s13660-017-1499-4

Qi, L., Luo, L.: Tensor Analysis: Spectral Theory and Special Tensors Paperback. SIAM, New Delhi (2017)

Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences, Vol. 9 of Classics in Applied Mathematics, SIAM, Philadelphia. Revised reprint of the 1979 original (1994)

Shao, J.Y.: A general product of tensors with applications. Linear Algebra Appl. 439(8), 2350–2366 (2013)

Ravi, B., Bapat Ravindra, B., Raghavan, T.: Nonnegative Matrices and Applications, vol. 64. Cambridge University Press, Cambridge (1997)

Bhatia, R.: Matrix Analysis. Springer, New York (1997)

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