  The spectral norm and the nuclear norm of a third order tensor play an
important role in the tensor completion and recovery problem. We show that the
spectral norm of a third order tensor is equal to the square root of the
spectral norm of three positive semi-definite biquadratic tensors, and the
square roots of the nuclear norms of those three positive semi-definite
biquadratic tensors are lower bounds of the nuclear norm of that third order
tensor. This provides a way to estimate and to evaluate the spectral norm and
the nuclear norm of that third order tensor. Some upper and lower bounds for
the spectral norm and nuclear norm of a third order tensor, by spectral radii
and nuclear norms of some symmetric matrices, are presented.
