Open AccessGeneralized Commuting Maps On The Set of Singular Matrices
Author(s) -
Willian Franca,
Nelson Louza
Publication year - 2019
Publication title -
the electronic journal of linear algebra
Language(s) - English
Resource type - Journals
eISSN - 1537-9582
pISSN - 1081-3810
DOI - 10.13001/ela.2019.5173
Subject(s) - mathematics , combinatorics , field (mathematics) , matrix (chemical analysis) , singular value , set (abstract data type) , pure mathematics , physics , chemistry , eigenvalues and eigenvectors , chromatography , quantum mechanics , computer science , programming language
Let Mn(K) be the ring of all n × n matrices over a field K. In the present paper, additive mappings G : Mn(K) → Mn(K) such that [[G(y), y], y] = 0 for all singular matrix y will be characterized. Precisely, it will be proved that G(x) = λx + µ(x) for all x ∈ Mn(K), where λ ∈ K and µ is a central map. As an application, the description is given of all additive maps g : Mn(K) → Mn(K) such that [[g(yk1 ), yk2 ], yk3 ] = 0 for all singular matrices y ∈ Mn(K), where m ∈ N∗.