  In this paper, we study the Coxeter transformation of the derived categories
of coherent sheaves on smooth complete varieties. We first obtain that if the
rank of the Grothendieck group is finite, say $m$, then its characteristic
polynomials is $(\lambda+(-1)^{n})^m$, where $n$ is dimension of the variety.
We then study the Jordan canonical forms of the Coxeter transformations for
rational surfaces, smooth complete toric varieties with ample canonical or
anticanonical bundles, and prove that their Jordan canonical forms can
determine and be determined by the Betti numbers of these varieties. As an
application, we compute the Jordan canonical forms of tensor products of
matrices.
