A power Cayley-Hamilton identity for n × n matrices over a Lie nilpotent ring of index k
| Autor: | Leon van Wyk, Jenő Szigeti, Szilvia Szilágyi |
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| Rok vydání: | 2020 |
| Předmět: |
Numerical Analysis
Algebra and Number Theory 010102 general mathematics 010103 numerical & computational mathematics 01 natural sciences Matrix ring Matrix (mathematics) Crystallography Nilpotent Discrete Mathematics and Combinatorics Geometry and Topology 0101 mathematics Quotient ring Characteristic polynomial Mathematics |
| Zdroj: | Linear Algebra and its Applications. 584:153-163 |
| ISSN: | 0024-3795 |
| Popis: | For an n × n matrix A over a Lie nilpotent ring R of index k, with k ≥ 2 , we prove that an invariant “power” Cayley-Hamilton identity ( I n λ 0 ( 2 ) + A λ 1 ( 2 ) + ⋯ + A n 2 − 1 λ n 2 − 1 ( 2 ) + A n 2 λ n 2 ( 2 ) ) 2 k − 2 = 0 of degree n 2 2 k − 2 holds. The right coefficients λ i ( 2 ) ∈ R , 0 ≤ i ≤ n 2 are not uniquely determined by A, and the cosets λ i ( 2 ) + D , with D the double commutator ideal R [ [ R , R ] , R ] R of R, appear in the so-called second right characteristic polynomial p A ‾ , 2 ( x ) of the natural image A ‾ of A in the n × n matrix ring M n ( R / D ) over the factor ring R / D : p A ‾ , 2 ( x ) = ( λ 0 ( 2 ) + D ) + ( λ 1 ( 2 ) + D ) x + ⋯ + ( λ n 2 − 1 ( 2 ) + D ) x n 2 − 1 + ( λ n 2 ( 2 ) + D ) x n 2 . |
| Databáze: | OpenAIRE |
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