The tensor power method generalizes the matrix power method to higher order arrays, or tensors. Like in the matrix case, the fixed points of the tensor power method are the eigenvectors of the tensor. While every real symmetric matrix has an eigendecomposition, the vectors generating a symmetric decomposition of a real symmetric tensor are not always eigenvectors of the tensor. In this paper we show that whenever an eigenvector is a generator of the symmetric decomposition of a symmetric tensor, then (if the order of the tensor is sufficiently high) this eigenvector is robust, i.e., it is an attracting fixed point of the tensor power method. We exhibit new classes of symmetric tensors whose symmetric decomposition consists of eigenvectors. Generalizing orthogonally decomposable tensors, we consider equiangular tight frame decomposable and equiangular set decomposable tensors. Our main result implies that such tensors can be decomposed using the tensor power method.

翻译：张量幂方法将矩阵幂方法推广至高阶数组，即张量。与矩阵情形类似，张量幂方法的不动点即为张量的特征向量。虽然每个实对称矩阵都具有特征分解，但生成实对称张量对称分解的向量并不总是张量的特征向量。本文证明，当某个特征向量是对称张量对称分解的生成元时（若张量阶数足够高），该特征向量具有鲁棒性，即成为张量幂方法的吸引不动点。我们展示了几类新的对称张量，其对称分解由特征向量构成。通过推广正交可分解张量，我们研究了等角紧框架可分解张量与等角集可分解张量。我们的主要结果表明，此类张量可利用张量幂方法进行分解。