Abstract
Let M ( n,K) be the algebra of n × n matrices over an algebraically closed field K and T: M ( n, K)→ M ( n, K) a linear transformation with the property that T maps nonsingular (singular) matrices to nonsingular (singular) matrices. Using some elementary facts from commutative algebra we show that T is nonsingular and maps singular matrices to singular matrices ( T is nonsingular or T maps all matrices to singular matrices). Using these results we obtain Marcus and Moyl's characterization [ T( x) = UXV or U t XV for fixed U and V] from a result of Dieudonné's. Examples are given to show the hypothesis of algebraic closure in necessary.
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