The computational function of a matchgate is represented by its character matrix. In this article, we show that all nonsingular character matrices are closed under matrix inverse operation, so that for every $k$, the nonsingular character matrices of
$k$-bit matchgates form a group, extending the recent work of Cai and Choudhary (2006) of the same result for the case of $k=2$, and that the single and the two-bit matchgates are universal for matchcircuits, answering a question of Valiant (2002).
We give a family of counter examples showing that the two sequences of polytopes $Phi_{n,n}$ and $Psi_{n,n}$ are different. These polytopes were defined recently by S. Friedland in an attempt at a polynomial time algorithm for graph isomorphism.
