McCoy, following Frobenius, studied a problem which can be described as follows. Let k be an arbitrary field, kc its algebraic closure, and
any algebra of n ⨯ n matrices over k which contains the identity I . Define a canonical ordering to be a set of n mappings λ of
, or of a subset
of
, into kc such that the sequence λ1(A),λ2(A), …, λn(A), for each A ∈
, consists of the characteristic values (roots of det(A — xI) = 0) of A, each with the right multiplicity. Define a canonical ordering to be a Frobenius ordering if, for all non-commutative polynomials f(x1, x2, … , xm) and all finite subsets A1, A2, …, Am of
,
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