  Biquadratic tensors play a central role in many areas of science. Examples
include elasticity tensor and Eshelby tensor in solid mechanics, and Riemann
curvature tensor in relativity theory. The singular values and spectral norm of
a general third order tensor are the square roots of the M-eigenvalues and
spectral norm of a biquadratic tensor. The tensor product operation is closed
for biquadratic tensors. All of these motivate us to study biquadratic tensors,
biquadratic decomposition and norms of biquadratic tensors. We show that the
spectral norm and nuclear norm for a biquadratic tensor may be computed by
using its biquadratic structure. Then, either the number of variables is
reduced, or the feasible region can be reduced. We show constructively that for
a biquadratic tensor, a biquadratic rank-one decomposition always exists, and
show that the biquadratic rank of a biquadratic tensor is preserved under an
independent biquadratic Tucker decomposition. We present a lower bound and an
upper bound of the nuclear norm of a biquadratic tensor. Finally, we define
invertible biquadratic tensors, and present a lower bound for the product of
the nuclear norms of an invertible biquadratic tensor and its inverse, and a
lower bound for the product of the nuclear norm of an invertible biquadratic
tensor, and the spectral norm of its inverse.
