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)
