A QR Algorithm for Symmetric Tensors

Abstract

We extend the celebrated QR algorithm for matrices to symmetric tensors. The algorithm, named QR algorithm for symmetric tensors (QRST), exhibits similar properties to its matrix version, and allows the derivation of a shifted implementation with faster convergence. We further show that multiple tensor eigenpairs can be found from a local permutation heuristic which is effectively a tensor similarity transform, resulting in the permuted version of QRST called PQRST. Examples demonstrate the remarkable effectiveness of the proposed schemes for finding stable and unstable eigenpairs not found by previous tensor power methods.