  Recent work on eigenvalues and eigenvectors for tensors of order m >= 3 has
been motivated by applications in blind source separation, magnetic resonance
imaging, molecular conformation, and more. In this paper, we consider methods
for computing real symmetric-tensor eigenpairs of the form Ax^{m-1} = \lambda x
subject to ||x||=1, which is closely related to optimal rank-1 approximation of
a symmetric tensor. Our contribution is a shifted symmetric higher-order power
method (SS-HOPM), which we show is guaranteed to converge to a tensor
eigenpair. SS-HOPM can be viewed as a generalization of the power iteration
method for matrices or of the symmetric higher-order power method.
Additionally, using fixed point analysis, we can characterize exactly which
eigenpairs can and cannot be found by the method. Numerical examples are
presented, including examples from an extension of the method to finding
complex eigenpairs.
