Rank-permutable additive mappings
Let ▫$\sigma$▫ be a fixed non-identical permutation on ▫$k$▫ elements. Additive bijections ▫$T$▫ on the matrix algebra ▫$M_n(\mathbb{F})$▫ over a field ▫$\mathbb{F}$▫ of characteristic zero, with the property that ▫$\rm{rk} (A_1...A_k) = \rm{rk} (A_{\sigma(1)}...A_{\sigma(k)})$▫ implies the same condition on the ▫$T$▫ images, are characterized. It is also shown that the surjectivity assumption can be relaxed, if this property is preserved in both directions.
Bodi ▫$\sigma$▫ netrivialna permutacija na ▫$k$▫ elementih. Klasificiramo vse aditivne bijekcije ▫$T:M_n(F)\to M_n(F)$▫, ki ohranjajo ▫$\sigma$▫-rang permutabilnost na algebri matrik s koeficienti iz komutativnega obsega ▫$F$▫ ničelne karakteristike. Natančneje: Čim urejena ▫$k$▫-terka matrik ▫$(A_1,..,A_k)$▫ ustreza pogoju ▫$\rm{rk}(A_1...A_k) = \rm{rk}(A_{\sigma(1)} ... A_{\sigma(k)})$▫ potem isto velja za preslikano ▫$k$▫-terko ▫$(T(A_1),..,T(A_k))$▫. Če se ▫$\sigma$▫-rang permutabilnost ohranja v obeh smereh, lahko predpostavko o bijektivnosti omilimo.
2006
2013-10-15 12:05:02
1033
mathematics, linearna algebra, matrix algebra, rank, permutation, additive preservers
matematika, linearna algebra, matrična algebra, aditivni ohranjevalci, rang, permutacija
r6
Anna A.
Alieva
70
Aleksandr Èmilevič
Guterman
70
Bojan
Kuzma
70
ISSN
2
0024-3795
UDK
4
511.643
COBISS.SI-ID
3
13949273
0
Predstavitvena datoteka
2013-10-15 12:05:02