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ВИДЕОТЕКА |
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Matrix Centralizers and their Applications А. Э. Гутерман |
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Аннотация: The talk is based on the works [1, 2, 3]. For a matrix $$ \mathcal C (A)=\{X\in M_n (\mathbb F) | AX=XA \} $$ is the set of all matrices commuting with For a set $$ \mathcal C (S)=\{X\in M_n (\mathbb F) | AX=XA \text{ for every } A \in S\}=\bigcap_{A \in S} \mathcal C(A) $$ is the intersection of centralizers of all its elements. Centralizers are important and useful both in fundamental and applied sciences. A non-scalar matrix We investigate and characterize minimal and maximal matrices over arbitrary fields. Our results are then applied to the theory of commuting graphs of matrix rings and to characterize commutativity preserving maps on matrices. [1] G. Dolinar, A.E. Guterman, B. Kuzma, P. Oblak, Commuting graphs and extremal centralizers, Ars Mathematica Contemporanea, 7(2), 2014, 453-459. [2] G. Dolinar, A.E. Guterman, B. Kuzma, P. Oblak, Commutativity preservers via matrix centralizers, Publicationes Mathematicae Debrecen, 84(3-4), 2014, 439–450. [3] G. Dolinar, A.E. Guterman, B. Kuzma, P. Oblak, Extremal matrix centralizers, Linear Algebra and its Applications, 438(7), 2013, 2904-2910. |