The solution of the Binet-Cauchy functional equation for square matrices

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Department of Mathematical Sciences


It is shown that if f{hook} : Mn(K)→K is a nonconstant solution of the Binet-Cauchy functional equation f{hook}(AB) = 1 n! ∑ |s|=n n sf{hook}(As)f{hook}(Bs) for A, B ∈ Mn(K) and if f{hook}(E) = 0 where E is the n × n matrix with all entries 1 n then f{hook} is given by f{hook}(A) = m(det A) where m is a multiplicative function on K. For f{hook}(E)≠0 it has been shown by Heuvers, Cummings and Bhaskara Rao, that f{hook}(A) = φ(per A) where φ is an isomorphism of K. Thus the Binet-Cauchy functional equation is the source of the common properties of det A and per A. The value of f{hook}(E) is sufficient to distinguish between the two functions.

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Discrete Mathematics