  We develop a higher order generalization of the LQ decomposition and show
that this decomposition plays an important role in likelihood-based estimation
and testing for separable, or Kronecker structured, covariance models, such as
the multilinear normal model. This role is analogous to that of the LQ
decomposition in likelihood inference for the multivariate normal model.
Additionally, this higher order LQ decomposition can be used to construct an
alternative version of the popular higher order singular value decomposition
for tensor-valued data. We also develop a novel generalization of the polar
decomposition to tensor-valued data.
