Computing tensor Z-eigenvalues via shifted inverse power method

Abstract

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 P or P0 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 P or P0 tensor and symmetric tensor Z-eigenproblems, some numerical results are reported to illustrate the effectiveness of the proposed method.