Geodesic
Linear Transformations On Matrices: The Invariance of Generalized Permutation Matrices, I
Canadian journal of mathematics, Tome 28 (1976) no. 3, pp. 455-472

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Let F be a field, Mn(F) be the vector space of all w-square matrices with entries in F and a subset of Mn(F). It is of interest to determine the structure of linear maps T : Mn(F) →Mn(F) such that . For example: Let be GL(n, C), the group of all nonsingular n X n matrices over C [5]; the subset of all rank 1 matrices in MmXn(F) [4] (MmXn(F) is the vector space of all m X n matrices over F) ; the unitary group [2] ; or the set of all matrices X in Mn(F) such that det(X) = 0 [1]. Other results in this direction can be found in [3]. 
Ong, Hock; Botta, E. P. Linear Transformations On Matrices: The Invariance of Generalized Permutation Matrices, I. Canadian journal of mathematics, Tome 28 (1976) no. 3, pp. 455-472. doi: 10.4153/CJM-1976-047-9
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[1] 1. Dieudonné, J., Sur une généralisation du groupe orthogonal a quatre variables, Arch. Math. 1 (1949), 282–287. Google Scholar

[2] 2. Marcus, M., All linear operators leaving the unitary group invariant, Duke Math. J. 26 (1959), 155–163. Google Scholar

[3] 3. Marcus, M., Linear transformations on matrices, J. Res. NBS 75B (Math. Sci.) No. 3 and 4 (1971), 107–113. Google Scholar

[4] 4. Marcus, M. and Moyls, B., Transformations on tensor product spaces, Pacific J. Math. 9 (1959), 1215–1221. Google Scholar

[5] 5. Marcus, M. and Purves, R., Linear transformations on algebras of matrices II: The invariance of the elementary symmetric functions, Can. J. Math. 11 (1959), 383–396. Google Scholar

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