  We study a collection of concepts and theorems that laid the foundation of
matchgate computation. This includes the signature theory of planar matchgates,
and the parallel theory of characters of not necessarily planar matchgates. Our
aim is to present a unified and, whenever possible, simplified account of this
challenging theory. Our results include: (1) A direct proof that Matchgate
Identities (MGI) are necessary and sufficient conditions for matchgate
signatures. This proof is self-contained and does not go through the character
theory. More importantly it rectifies a gap in the existing proof. (2) A proof
that Matchgate Identities already imply the Parity Condition. (3) A simplified
construction of a crossover gadget. This is used in the proof of sufficiency of
MGI for matchgate signatures. This is also used to give a proof of equivalence
between the signature theory and the character theory which permits omittable
nodes. (4) A direct construction of matchgates realizing all
matchgate-realizable symmetric signatures.

Computational Complexity