For a given (terminating) term rewriting system

one can often estimate its \emph{derivational complexity} indirectly

by looking at the proof method that established termination. In this

spirit we investigate two instances of the interpretation method:

\emph{matrix interpretations} and \emph{context dependent interpretations}.

We introduce a subclass of matrix interpretations, denoted

as \emph{triangular matrix interpretations}, which induce

polynomial derivational complexity and establish tight correspondence

results between a subclass of context dependent interpretations and

restricted triangular matrix interpretations.

The thus obtained new results are easy to implement and

considerably extend the analytic power of existing results.

We provide ample numerical data for assessing the viability of the method.