Graphical evaluation of sparse determinants


Abstract. In this paper, we show that the determinant of a matrix with entries in a commutative ring can be recursively computed by use of an associated directed graph whose circuits are assigned weights in the ring. This result provides an efficient means of calculating the determinants of sparse matrices. As an application, we compute the determinants of the Cartan matrices associated to the simple complex Lie algebras. Let M = (m¡j) be an « X « matrix with entries in a commutative ring A with identity element 1 different from 0. Define the associated graph of M to be the directed graph T with n vertices (numbered from 1 to n) which has a directed edge (i,j) from vertex i to vertex j precisely when mtj # 0 (/ ' =f is allowed). A circuit of length k (1 < k < n) in T is a /c-cycle2 y = [«„..., ik] such that (ir, ir+,) (for 1 < r < k) and (ik, i,) are directed edges of T. The ring element w(y) = ( — l)k~xm ¡ , • • • m., m ¡ ¡ (w(y) = m¡, if k = 1) is called the weight of y. We let |A/(y) | denote the determinant of the submatrix M(y) obtained by deleting the rows and columns of M numbered /,,..., ik. (When k = n, we set |Af(y) | = 1.) For each vertex i, let C(i) denote the set of all circuits to which i belongs. Theorem. Proof. of \M \ is For any vertex i, \M \ = 2yeC(0 w(y)\M(y)\. Assume without loss of generality that / = 1. The usual expansion \M \ = 2 (sgn o)mXMX)m2M2) • • • mna{n) where the sum ranges over all permutations a of (1, 2,..., n). Without affecting the result, we can restrict the sum to those a for which ml,o<l)Wo(l),a2(l) • • • ™<,<-'(l),l ^ 0> where k is the smallest positive integer such that ak(\) = 1. In terms of the associated graph T, this restricts the sum to the permutations a for which the k-cycle [1, a(l),..., ak~x(l)] is a circuit of T with nonzero weight. Summing over all a corresponding to a single circuit y with w(y) = £ 0 yields the partia

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