  Let F be a 4-regular graph with an Euler system C. We introduce a simple way
to modify the interlacement matrix of C so that every circuit partition P of F
has an associated modified interlacement matrix M(C,P). If C and C' are Euler
systems of F then M(C,C') and M(C',C) are inverses, and for any circuit
partition P, M(C',P)=M(C',C)M(C,P). This machinery allows for short proofs of
several results regarding the linear algebra of interlacement.
