Applications of graph theory to matrix theory

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Abstract

Let A l Ak be n x n matrices over a commutative ring R with identity. Graph theoretic methods are established to compute the standard polynomial [A , ... I Ak]. It is proved that if k < 2n 2, and if the characteristic of R either is zero or does not divide 4I(V2 n) 2, where I denotes the greatest integer function, then there exist n x n skew-symmetric matrices A 1 . . . , Ak such that [A 1, . . . AkI AO.