The positive definiteness of an even degree homogeneous polynomial plays an important role in the stability study of nonlinear autonomous systems via Lyapunov’s direct method in automatic control, the detection of $\mathcal{P}$ or ${\mathcal{P}}_0$ tensor in tensor complementarity problems and spectral hypergraph theory, and more. Owing to the positive definiteness of an even degree homogeneous polynomial is equivalent to that of an even order symmetric tensor. In this paper, we propose a shifted inverse power method for computing tensor Z-eigenpairs, which can be viewed as a generalization of the inverse power method for matrices case. We also formulate it as a fixed point iteration form, and reveal that the relationship between the fixed points and the Z-eigenvectors of symmetric tensors. The advantages of the proposed method are simple operations and readily comprehensible convergence analysis. An efficient initialization strategy is also developed, which makes the proposed method converges to a better solution compared to not using the initialization strategy. Finally, we present applications of the proposed method in nonlinear autonomous systems, the detection of $\mathcal{P}$ or ${\mathcal{P}}_0$ tensor and symmetric tensor Z-eigenproblems, some numerical results are reported to illustrate the effectiveness of the proposed method.

偶次齐次多项式的正定性在非线性自治系统的稳定性研究中起着重要的作用，通过李雅普诺夫直接法在自动控制中，检测 $\mathcal{磷}$ 或者 ${\mathcal{磷}}_0$张量互补问题和谱超图理论中的张量等等。由于偶次齐次多项式的正定性等价于偶次对称张量的正定性。在本文中，我们提出了一种用于计算张量 Z 特征对的平移逆幂方法，可以将其视为矩阵情况下逆幂方法的推广。我们还将其表述为不动点迭代形式，并揭示了不动点与对称张量的 Z 特征向量之间的关系。该方法的优点是操作简单，收敛分析容易理解。还开发了一种有效的初始化策略，与不使用初始化策略相比，这使得所提出的方法收敛到更好的解决方案。最后，$\mathcal{磷}$ 或者 ${\mathcal{磷}}_0$ 张量和对称张量 Z 特征问题，报告了一些数值结果来说明所提出方法的有效性。