Classification of pairs consisting of a linear and a semilinear map |
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Authors: | Dragomir Ž Djoković |
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Institution: | Department of Pure Mathematics University of Waterloo Waterloo, Ontario, Canada, N2L 3G1 |
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Abstract: | L. Kronecker has found normal forms for pairs (A, B) of m-by-n matrices over a field F when the admissible transformations are of the type (A, B)→(SAT, SBT), where S and T are invertible matrices over F. For the details about these normal forms we refer to Gantmacher's book on matrices 5, Chapter XII]. See also Dickson's paper 3]. We treat here the following more general problem: Find the normal forms for pairs (A, B) of m-by-n matrices over a division ring D if the admissible transformations are of the type (A, B)→(SAT, SBJ(T)) where J is an automorphism of D. It is surprising that these normal forms (see Theorem 1) are as simple as in the classical case treated by Kronecker. The special case D=C, J=conjugation is essentially equivalent to the recent problem of Dlab and Ringel 4]. This is explained thoroughly in Sec. 6. We conclude with two open problems. |
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