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Positive Definiteness of a Class of Cyclic Symmetric Tensors

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  • Yisheng Song

    (Chongqing Normal University)

Abstract

For a fourth order three-dimensional cyclic symmetric tensor, a sufficient and necessary condition is bulit for its positive semi-definiteness. A sufficient and necessary condition of positive definiteness is showed for a fourth order n-dimensional symmetric tensor. With the help of such a condition, the positive definiteness of a class of fourth order three-dimensional cyclic symmetric tensors is given. Moreover, the positive definiteness of a class of non-cyclic symmetric tensors is showed also. By applying these conclusions, several (strict) inequalities are erected for ternary quartic homogeneous polynomials.

Suggested Citation

  • Yisheng Song, 2025. "Positive Definiteness of a Class of Cyclic Symmetric Tensors," Journal of Optimization Theory and Applications, Springer, vol. 204(3), pages 1-19, March.
    • Handle: RePEc:spr:joptap:v:204:y:2025:i:3:d:10.1007_s10957-025-02615-6
    DOI: 10.1007/s10957-025-02615-6
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    References listed on IDEAS

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    1. Yisheng Song & Liqun Qi, 2016. "Eigenvalue analysis of constrained minimization problem for homogeneous polynomial," Journal of Global Optimization, Springer, vol. 64(3), pages 563-575, March.
    2. Shenglong Hu & Guoyin Li & Liqun Qi & Yisheng Song, 2013. "Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming," Journal of Optimization Theory and Applications, Springer, vol. 158(3), pages 717-738, September.
    3. Qin Ni & Liqun Qi, 2015. "A quadratically convergent algorithm for finding the largest eigenvalue of a nonnegative homogeneous polynomial map," Journal of Global Optimization, Springer, vol. 61(4), pages 627-641, April.
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