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Linear preservers on matrices. (English) Zbl 0619.15003

Let \(M_ n(F)\) denote the set of \(n\times n\) matrices with entries in a field F. Linear maps \(L: M_ n(F)\to M_ n(F)\) are considered such that L(adj A)\(=adj(L(A))\) for all \(A\in M_ n(F)\) where F is infinite. For F equal to either the real or complex numbers linear maps are considered which preserve the idempotent matrices and satisfy \(L(I_ n)=I_ n\). From this last result the linear maps which satisfy \(L(e^ A)=e^{L(A)}\) for all \(A\in M_ n(F)\) are characterized.
Reviewer: R.Sinkhorn

MSC:

15A04 Linear transformations, semilinear transformations
15A27 Commutativity of matrices
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References:

[1] Djokovíc, D.Ž., Linear transformations of tensor products preserving a fixed rank, Pacific J. Math, 30, 411-414 (1969)
[2] Frobenius, G., Über die Darstellung der endlichen Gruppen durch lineare Substitutionen: I, Sitzungsber. Preuss. Akad. Wiss. Berlin, 994-1015 (1897) · JFM 28.0130.01
[3] Gantmacher, F. R., The Theory of Matrices, Vol. 2 (1959), Chelsea: Chelsea New York · Zbl 0085.01001
[4] Howard, R., Linear maps that preserve matrices annihilated by a polynomial, Linear Algebra Appl., 30, 167-176 (1980) · Zbl 0439.15005
[5] Hua, L. K., A theorem on matrices over a field and its applications, Acta Math. Sinica, 1, 109-163 (1951)
[6] Lim, M. H., Rank \(k\) preservers on second Grassmann product spaces, Malaysian J. Sci., 3B, 145-149 (1975)
[7] Lim, M. H., Linear transformations on symmetric matrices, Linear and Multilinear Algebra, 7, 45-57 (1979) · Zbl 0498.15008
[8] Lim, M. H., Linear mappings on symmetry classes of tensors, Linear and Multilinear Algebra, 12, 109-123 (1982) · Zbl 0494.15015
[9] Marcus, M.; Moyls, B., Transformations on tensor product spaces, Pacific J. Math., 9, 1215-1221 (1959) · Zbl 0089.08902
[10] Marcus, M.; Purves, R., Linear transformations on algebras of matrices II: The invariance of the elementary symmetric functions, Canad. J. Math., 11, 383-396 (1959) · Zbl 0086.01704
[11] Marcus, M.; Westwick, R., Linear maps on skew-symmetric matrices: The invariance of elementary symmetric functions, Pacific J. Math., 10, 917-924 (1960) · Zbl 0093.24204
[12] Pierce, S., Linear operators preserving the real symplectic group, Canad. J. Math., 27, 715-724 (1975) · Zbl 0284.15003
[13] Sinkhorn, R., Linear adjugate preservers on the complex matrices, Linear and Multilinear Algebra, 12, 215-222 (1982) · Zbl 0496.15003
[14] Westwick, R., Linear transformations on Grassmann spaces III, Linear and Multilinear Algebra, 2, 257-268 (1974) · Zbl 0297.15026
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