Let R denote a commutative ring with identity and (R) denote the
ring of n* n matrices over R . For 90 years much effor t has been devoted
to the following question. Suppose P(A) is an invariant defined on matrices
A in (R) . Then, determine the set of R-linear mappings T : (R) -*• (R)
which preserve the invariant, i . e . , find the linear mappings T satisfying
p(T(A)) = P(A) for all A in (R) . For example, the invariant might be
the determinant, i . e . , p(A) = det(A) , and here we would seek all linear
mappings T with det(T(A)] = det(A) .
The case of the determinant was f i r s t studied where R = £ , the complex
numbers, by Frobenius in 1897 [1]. Frobenius showed that i f T : ((£) •+ (CC)
preserved determinants, then either T(A) = PAQ for all A in (C) or that
T(A) = PA Q for al l A in (C) where P and Q are invertible matrices
with det(PQ) = 1 .
Many of the questions regarding linear maps T : (R)n - (R) which
preserve invariants can be reduced to the problem of determining the set of
linear maps which carry the rank one matrices into themselves. This fact was
noted by M. Marcus in a survey articl e [10] in 1971 on this subject where R
was assumed to be a f i e l d . In 1959, assuming that k is an algebraically
closed fiel d of characteristic zero, then Marcus and Moyls [12] proved that,
i f T : ( k ) "
( k )
is a k-linear mapping with the property that
rank(T(A)) = 1 whenever rank(A) = 1 , then T has the form T(A) = PAQ for
all A in (k) or T(A) = PA^ for all A in (k) where P and Q are
invertible matrices. I f one examines carefully the proofs of Marcus and Moyls
[12], one finds that their proof carries over to. any hloJLd o& any ckaKao£vva-
tic. The Marcus and Moyls result was proven by the use of multilinear algebra.
An elementary matrix theoretic proof of the same result was given by Mine [17]
Received by Editor Dec. 7, 1981
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