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Linear operators preserving idempotent matrices over fields. (English) Zbl 0718.15004

If F is a field of characteristic \(p\neq 2\), then the semigroup \({\mathcal F}(F)\) of linear operators on the \(n\times n\) matrices over F that preserve idempotence is generated by a transposition and the similarity operators. An operator strongly preserves idempotence if it preserves both idempotence and nonidempotence. Over fields of characteristic \(p\neq 2\), this is equivalent to preserving idempotence. The structure of the semigroup \({\mathcal L}(F)\) of linear operators that strongly preserve idempotence is described in the case of characteristic two.

MSC:

15A04 Linear transformations, semilinear transformations
15A30 Algebraic systems of matrices
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References:

[1] L.B. Beasley and N.J. Pullman, Linear operators strongly preserving idempotent matrices over semirings, to appear.; L.B. Beasley and N.J. Pullman, Linear operators strongly preserving idempotent matrices over semirings, to appear. · Zbl 0744.15010
[2] Chan, G. H.; Lim, M. H.; Tan, K. K., Linear preservers on matrices, Linear Algebra Appl., 93, 67-80 (1987) · Zbl 0619.15003
[3] Howard, R., Linear maps that preserve matrices annihilated by a polynomial, Linear Algebra Appl., 30, 167-176 (1980) · Zbl 0439.15005
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