A graph theoretic proof of the fundamental trace identity

Discrete Mathematics 69 (1988) W-198

197

, UnSvemi~~degli Studi di Lace, I-731

Received 26 March 1986 Revised 15 May 1987 To each circuit decomposition of a directed multigraph a permutation of the edge set is assigned, and a lemma on the signs of the resulting permutations is proved yielding the fundamental trace identity for matrices over a commutative unitary ring as a corollary.

Let r be a directed multigraph. A circuit decomposition of r is a circuits of r such that each edge of r occurs in exactly one element of circuit [yl, . . . , yJ E D determines a permutation of (yl, D. . , yt}, viz. (Y lr-•-9

r

y,

=

(

Yl,

l

l

Y20 * l

l

0

l

0

Yr-1, Yt

yr, y1 >’

and 00 := [YI,-..,Y,lED

(Y1,-*-P Yt)

is a permutation of the edge set of P: Let Z(r) := {a decomposition

! t.~= oD

for some circuit

,. Ilf r has more edges than vertices, then the number of’even permutations in %(lJ is equal to the number of odd permutations in 3(r).

at the same vertex. Let A be

Hartmut Laue

198

over R,

matrices

where (il i, . . . , iJ ranges over the set of all cycles of the cycle deco.mposition of the permutation 0. e

put

1:=

Ch,...A)

is a k-fold R-linear mapping. Therefore we may assume tha hose only non-zero entry is a 1, say, in the (5, +)-p&e. irected multigraph F with vertex set (1, . . . , n} and with k Idges yl, . . . , yk s at 5 and ends at sj. Obviously, ,) # 0, i.e., Sii = rij+l for 1 s j C t and Sit= riI, for all cycles . in the cycle de mposition of 0. is is eq+Aent to (119 . . . ) iJ o&ring CI= aD for some circuit decomposition of K Therefore,

, . . . , lb&) =

-w?

sgn(a) = 0,

e assertion of our Corollary plays a fundamental role in the theory of trace aph theoretic approach in t note is closely related sur-Levitzky identity [1, I, eorem 141.

ollob&, Grap’nTheory (Springer, Berlin, 1979). Lew, The generalized Cayley-Hamilton theorem in p1dimensions, 2. Angw. Math. Phys. 17 si, The invariant theory of n X n m si, A formal inverse to the Cayley

Math. 19 (1976) 306-381. em, J. Algebra 107 (1987) 63-74.