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Linear transformations on algebras of matrices over the class of infinite fields Oishi, Tony Tsutomu

Abstract

The problem of determining the structure of linear transformations on the algebra of n-square matrices over the complex field is discussed by M. Marcus and B. N. Moyls in the paper ''Linear Transformations on Algebras of Matrices". The authors were able to characterize linear transformations which preserve one or more of the following properties of n-square matrices; rank, determinant and eigenvalues. The problem of obtaining a similar characterization of transformations as given by M. Marcus and B. N. Moyls but for a wider class of fields is considered in this thesis. In particular, their characterization of rank preserving transformations holds for an arbitrary field. One of the results on determinant preserving transformations obtained by M. Marcus and B. N. Moyls states that if a linear transformation T maps unimodular matrices into unimodular matrices, then T preserves determinants. Since this result does not necessarily hold for algebras of matrices over finite fields, the discussion on the characterization of determinant preserving transformations is limited to algebras of matrices over infinite fields.

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