# Cayley-Hamilton Theorem for Matrices over an Arbitrary Ring

• Volume: 32, Issue: 2-3, page 269-276
• ISSN: 1310-6600

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## Abstract

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2000 Mathematics Subject Classification: 15A15, 15A24, 15A33, 16S50.For an n×n matrix A over an arbitrary unitary ring R, we obtain the following Cayley-Hamilton identity with right matrix coefficients: (λ0I+C0)+A(λ1I+C1)+… +An-1(λn-1I+Cn-1)+An (n!I+Cn) = 0, where λ0+λ1x+…+λn-1 xn-1+n!xn is the right characteristic polynomial of A in R[x], I ∈ Mn(R) is the identity matrix and the entries of the n×n matrices Ci, 0 ≤ i ≤ n are in [R,R]. If R is commutative, then C0 = C1 = … = Cn-1 = Cn = 0 and our identity gives the n! times scalar multiple of the classical Cayley-Hamilton identity for A.

## How to cite

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Szigeti, Jeno. "Cayley-Hamilton Theorem for Matrices over an Arbitrary Ring." Serdica Mathematical Journal 32.2-3 (2006): 269-276. <http://eudml.org/doc/281373>.

@article{Szigeti2006,
abstract = {2000 Mathematics Subject Classification: 15A15, 15A24, 15A33, 16S50.For an n×n matrix A over an arbitrary unitary ring R, we obtain the following Cayley-Hamilton identity with right matrix coefficients: (λ0I+C0)+A(λ1I+C1)+… +An-1(λn-1I+Cn-1)+An (n!I+Cn) = 0, where λ0+λ1x+…+λn-1 xn-1+n!xn is the right characteristic polynomial of A in R[x], I ∈ Mn(R) is the identity matrix and the entries of the n×n matrices Ci, 0 ≤ i ≤ n are in [R,R]. If R is commutative, then C0 = C1 = … = Cn-1 = Cn = 0 and our identity gives the n! times scalar multiple of the classical Cayley-Hamilton identity for A.},
author = {Szigeti, Jeno},
journal = {Serdica Mathematical Journal},
keywords = {Commutator Subgroup [R,R] of a Ring R; Cayley-Hamilton theorem; preadjoint of matrix; right characteristic polynomial of matrix; commutators of rings},
language = {eng},
number = {2-3},
pages = {269-276},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Cayley-Hamilton Theorem for Matrices over an Arbitrary Ring},
url = {http://eudml.org/doc/281373},
volume = {32},
year = {2006},
}

TY - JOUR
AU - Szigeti, Jeno
TI - Cayley-Hamilton Theorem for Matrices over an Arbitrary Ring
JO - Serdica Mathematical Journal
PY - 2006
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 32
IS - 2-3
SP - 269
EP - 276
AB - 2000 Mathematics Subject Classification: 15A15, 15A24, 15A33, 16S50.For an n×n matrix A over an arbitrary unitary ring R, we obtain the following Cayley-Hamilton identity with right matrix coefficients: (λ0I+C0)+A(λ1I+C1)+… +An-1(λn-1I+Cn-1)+An (n!I+Cn) = 0, where λ0+λ1x+…+λn-1 xn-1+n!xn is the right characteristic polynomial of A in R[x], I ∈ Mn(R) is the identity matrix and the entries of the n×n matrices Ci, 0 ≤ i ≤ n are in [R,R]. If R is commutative, then C0 = C1 = … = Cn-1 = Cn = 0 and our identity gives the n! times scalar multiple of the classical Cayley-Hamilton identity for A.
LA - eng
KW - Commutator Subgroup [R,R] of a Ring R; Cayley-Hamilton theorem; preadjoint of matrix; right characteristic polynomial of matrix; commutators of rings
UR - http://eudml.org/doc/281373
ER -

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