  We study the entanglement in tripartite quantum systems by using the
principal basis matrix representations of density matrices. Using the Schmidt
decomposition and local unitary transformation, we first convert the general
states to simpler forms and then construct some special matrices from the
correlation tensors of the simplified density matrices. Based on the different
linear combinations of these matrices, necessary conditions are presented to
detect entanglement of tripartite states. Detailed examples show that our
method can detect more entangled states than previous ones.
