Linear commutativity preservers

Linearni ohranjevalci komutativnosti na matrični algebri so tesno povezani s preslikavami, ki vse matrike ranga 1 preslikajo v matrike ranga 1. Zato najprej določimo obliko linearnih preslikav, ki ohranjajo rang 1, nato pa natančno opišemo nesingularne linearne preslikave na algebri kvadratnih matrik z elementi iz algebraično zaprtega polja F s karakteristiko 0. Z močnejšo predpostavko, da preslikava ohranja komutativnost v obe smeri, vendar sedaj brez predpostavke surjektivnosti, dobimo podoben rezultat. Linearne preslikave, ki ohranjajo komutativnost oziroma ohranjajo komutativnost v obe smeri, proučimo tudi na algebri zgoraj trikotnih matrik nad poljubnim poljem in na realni (jordanski) algebri hermitskih kompleksnih matrik. V slednjem primeru dobimo karakterizacijo celo brez predpostavke surjektivnosti in z ohranjanjem komutativnosti samo v eno smer.Linear commutativity-preservers on matrix algebra are linked to linear maps which map every rank-one matrix to a rank-one matrix. So, firstly we develope the structure of linear maps that preserve rank 1. After that we describe all non-singular linear maps on the algebra of all square matrices over an algebraically closed field F with characteristic 0. Furthermore, by modifying the hypothesis and leaving out the non-singularity, we arrive to the similar conclusion. In this case, the stronger assumption is needed, which is preserving commutativity in both directions. We also consider linear maps that preserve commutativity in one or in both directions on the algebra of all upper-triangular matrices over any field and on the real (jordan) algebra of all complex self-adjoint matrices. In second case, the characterization is obtained without non-singularity assumption and preserving commutativity in one direction only