US- and U-Eigenpairs of Complex Tensors

In this chapter we discuss the computation of US-eigenpairs of complex symmetric tensors and U-eigenpairs of complex tensors. We derive an iterative algorithm for computing US-eigenpairs of complex symmetric tensors which is based on the Takagi factorization of complex symmetric matrices that are denoted as the QRCST Algorithm. We observe that multiple US-eigenpairs can be found from a local permutation heuristic, which is effectively a tensor similarity transformation, resulting in the permuted version of QRCST. We also present a generalization of their techniques to general complex tensors and derive a higher-order power type method for computing a US- or a U-eigenpair, similar to the higher-order power method for computing Z-eigenpairs of real symmetric tensors or a best rank-one approximation of real tensors. We illustrate the algorithms via numerical examples.