  In this paper, it is proved that (strict) copositivity of a symmetric tensor
$\mathcal{A}$ is equivalent to the fact that every principal sub-tensor of
$\mathcal{A}$ has no a (non-positive) negative $H^{++}$-eigenvalue. The
necessary and sufficient conditions are also given in terms of the
$Z^{++}$-eigenvalue of the principal sub-tensor of the given tensor. This
presents a method of testing (strict) copositivity of a symmetric tensor by
means of the lower dimensional tensors. Also the equivalent definition of
strictly copositive tensors is given on entire space $\mathbb{R}^n$.
